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PURE AND APPLIED MATHEMATICS 

A Series of Texts and Monographs 

Edited by: R. COURANT ■ L. BERS • J. J. STOKER 



Vol. I: Supersonic Flow and Shock Waves 

By R. Courant and K. O. Friedrichs 

Vol. II: Non-linear Vibrations in Mechanical and Electrical 
Systems 
By J. J. Stoker 

Vol. Ill: Dirichlet's Principle, Conformal Mapping, and 
Minimal Surfaces 

By R. Courant 

Vol. IV: Water Waves 

By J. J. Stoker 

Vol. V: Integral Equations 

By F. G. Tricomi 

Vol. VI: Differential Equations: Geometric Theory 

By Solomon Lefschetz 

Vol. VII: Linear Operators — Parts I and II 

By Nelson Dunford and Jacob T. Schwartz 

Vol. VIII: Modern Geometrical Optics 

By Max Herzberger 

Vol. IX: Orthogonal Expansions 

By G. Sansone 

Vol. X: Lectures on Differential and Integral Equations 

By K. Yosida 

Vol. XI: Representation Theory of Finite Groups and 
Associative Algebras 

By C. W. Curtis and I. Reiner 

Vol. XII: Electromagnetic Theory and Geometrical Optics 

By Morris Kline and Irvin W. Kay 

Additional volumes in preparation 



PURE AND APPLIED MATHEMATICS 

A Series of Texts and Monographs 

Edited by 
R. COURANT L. BERS ■ J. J. STOKER 



VOLUME IV 



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Waves about a harbor 



WATER WAVES 

The Mathematical Theory 
with Applications 



J. J. STOKER 

INSTITUTE OF MATHEMATICAL SCIENCES 
NEW YORK UNIVERSITY, NEW YORK 



INTERSCIENCE PUBLISHERS, INC., NEW YORK 



All Rights Reserved 

Library of Congress Catalog Card Number 
56-8228 



Copyright © 1957 
7 8 9 10 



ISBN 470 82863 3 

PRINTED IN THE UNITED STATES OF AMERICA 



To 
NANCY 



Introduction 



1. Introduction 



The purpose of this book is to present a connected account of the 
mathematical theory of wave motion in liquids with a free surface 
and subjected to gravitational and other forces, together with ap- 
plications to a wide variety of concrete physical problems. 

Surface wave problems have interested a considerable number of 
mathematicians beginning apparently with Lagrange, and con- 
tinuing with Cauchy and Poisson in France.* Later the British school 
of mathematical physicists gave the problems a good deal of atten- 
tion, and notable contributions were made by Airy, Stokes, Kelvin, 
Rayleigh, and Lamb, to mention only some of the better known. In 
the latter part of the nineteenth century the French once more took 
up the subject vigorously, and the work done by St. Venant and 
Boussinesq in this field has had a lasting effect: to this day the 
French have remained active and successful in the field, and par- 
ticularly in that part of it which might be called mathematical 
hydraulics. Later, Poincare made outstanding contributions par- 
ticularly with regard to figures of equilibrium of rotating and gravi- 
tating liquids (a subject which will not be discussed in this book); 
in this same field notable contributions were made even earlier 
by Liapounoff. One of the most outstanding accomplishments in the 
field from the purely mathematical point of view — the proof of the 
existence of progressing waves of finite amplitude — was made by 
Nekrassov [N.l], [N.la]f in 1921 and independently by a different 
means by Levi-Civita [L.7] in 1925. 

The literature concerning surface waves in water is very extensive. 
In addition to a host of memoirs and papers in the scientific journals, 
there are a number of books which deal with the subject at length. 
First and foremost, of course, is the book of Lamb [L.3], almost 
a third of which is concerned with gravity wave problems. There 
are books by Bouasse [B.15], Thorade [T.4], and Sverdrup [S.39] 

* This list would be considerably extended (to include Euler, the Bernoullis, 

and others) if hydrostatics were to be regarded as an essential part of our subject. 

f Numbers in square brackets refer to the bibliography at the end of the book . 



X INTRODUCTION 

devoted exclusively to the subject. The book by Thorade consists 
almost entirely of relatively brief reviews of the literature up to 
1931 — an indication of the extent and volume of the literature 
on the subject. The book by Sverdrup was written with the special 
needs of oceanographers in mind. One of the main purposes of the 
present book is to treat some of the more recent additions to our 
knowledge in the field of surface wave problems. In fact, a large part 
of the book deals with problems the solutions of which have been 
found during and since World War II; this material is not available 
in the books just now mentioned. 

The subject of surface gravity waves has great variety whether 
regarded from the point of view of the types of physical problems 
which occur, or from the point of view of the mathematical ideas 
and methods needed to attack them. The physical problems range 
from discussion of wave motion over sloping beaches to flood waves 
in rivers, the motion of ships in a sea-way, free oscillations of enclosed 
bodies of water such as lakes and harbors, and the propagation of 
frontal discontinuities in the atmosphere, to mention just a few. 
The mathematical tools employed comprise just about the whole of 
the tools developed in the classical linear mathematical physics 
concerned with partial differential equations, as well as a good part 
of what has been learned about the nonlinear problems of mathe- 
matical physics. Thus potential theory and the theory of the linear 
wave equation, together with such tools as conformal mapping and 
complex variable methods in general, the Laplace and Fourier 
transform techniques, methods employing a Green's function, integral 
equations, etc. are used. The nonlinear problems are of both elliptic 
and hyperbolic type. 

In spite of the diversity of the material, the book is not a collection 
of disconnected topics, written for specialists, and lacking unity and 
coherence. Instead, considerable pains have been taken to supply 
the fundamental background in hydrodynamics — and also in some 
of the mathematics needed — and to plan the book in order that it 
should be as much as possible a self-contained and readable whole. 
Though the contents of the book are outlined in detail below, it has 
some point to indicate briefly here its general plan. There are four 
main parts of the book: 

Part I, comprising Chapters 1 and 2, presents the derivation of 
the basic hydrodynamic theory for non-viscous incompressible fluids, 
and also describes the two principal approximate theories which form 



INTRODUCTION XI 

the basis upon which most of the remainder of the book is built. 

Part II, made up of Chapters 3 to 9 inclusive, is based on the ap- 
proximate theory which results when the amplitude of the wave 
motions considered is small. The result is a linear theory which from 
the mathematical point of view is a highly interesting chapter in 
potential theory. On the physical side the problems treated include 
the propagation of waves from storms at sea, waves on sloping 
beaches, diffraction of waves around a breakwater, waves on a 
running stream, the motion of ships as floating rigid bodies in a sea- 
way. Although this theory was known to Lagrange, it is often referred 
to as the Cauchy-Poisson theory, perhaps because these two mathe- 
maticians were the first to solve interesting problems by using it. 

Part III, made up of Chapters 10 and 11, is concerned with problems 
involving waves in shallow water. The approximate theory which 
results from assuming the water to be shallow is not a linear theory, 
and wave motions with amplitudes which are not necessarily small 
can be studied by its aid. The theory is often attributed to Stokes 
and Airy, but was really known to Lagrange. If linearized by making 
the additional assumption that the wave amplitudes are small, the 
theory becomes the same as that employed as the mathematical 
basis for the theory of the tides in the oceans. In the lowest order 
of approximation the nonlinear shallow water theory results in a 
system of hyperbolic partial differential equations, which in im- 
portant special cases can be treated in a most illuminating way with 
the aid of the method of characteristics. The mathematical methods 
are treated in detail in Chapter 10. The physical problems treated in 
Chapter 10 are quite varied; they include the propagation of unsteady 
waves due to local disturbances into still water, the breaking of 
waves, the solitary wave, floating breakwaters in shallow water. A 
lengthy section on the motions of frontal discontinuities in the 
atmosphere is included also in Chapter 10. In Chapter 11, entitled 
Mathematical Hydraulics, the shallow water theory is employed to 
study wave motions in rivers and other open channels which, unlike 
the problems of the preceding chapter, are largely conditioned by 
the necessity to consider resistances to the flow due to the rough 
sides and bottom of the channel. Steady flows, and steady progressing 
waves, including the problem of roll waves in steep channels, are 
first studied. This is followed by a treatment of numerical methods 
of solving problems concerning flood-waves in rivers, with the object 
of making flood predictions through the use of modern high speed 



XII INTRODUCTION 

digital computers. That such methods can be used to furnish accurate 
predictions has been verified for a flood in a 400-mile stretch of the 
Ohio River, and for a flood coming down the Ohio River and passing 
through its junction with the Mississippi River. 

Part IV, consisting of Chapter 12, is concerned with problems 
solved in terms of the exact theory, in particular, with the use of the 
exact nonlinear free surface conditions. A proof of the existence of 
periodic waves of finite amplitude, following Levi-Civita in a general 
way, is included. 

The amount of mathematical knowledge needed to read the book 
varies in different parts. For considerable portions of Part II the 
elements of the theory of functions of a complex variable are assumed 
known, together with some of the standard facts in potential theory. 
On the other hand Part III requires much less in the way of specific 
knowledge, and, as was mentioned above, the basic theory of the 
hyperbolic differential equations used there is developed in all detail 
in the hope that this part would thus be made accessible to engineers, 
for example, who have an interest in the mathematical treatment of 
problems concerning flows and wave motions in open channels. 

In general, the author has made considerable efforts to try to 
achieve a reasonable balance between the mathematics and the 
mechanics of the problems treated. Usually a discussion of the physical 
factors and of the reasons for making simplified assumptions in each 
new type of concrete problem precedes the precise formulation of the 
mathematical problems. On the other hand, it is hoped that a clear 
distinction between physical assumptions and mathematical deduc- 
tions — so often shadowy and vague in the literature concerned 
with the mechanics of continuous media — has always been main- 
tained. Efforts also have been made to present important portions 
of the book in such a way that they can be read to a large extent 
independently of the rest of the book; this was done in some cases 
at the expense of a certain amount of repetition, but it seemed to 
the author more reasonable to save the time and efforts of the reader 
than to save paper. Thus the portion of Chapter 10 concerned with 
the dynamics of the motion of fronts in meteorology is largely 
self-contained. The same is true of Chapter 11 on mathematical 
hydraulics, and of Chapter 9 on the motion of ships. 

Originally this book had been planned as a brief general introduc- 
tion to the subject, but in the course of writing it many gaps and 
inadequacies in the literature were noticed and some of them have 



INTRODUCTION XIII 

been filled in; thus a fair share of the material presented represents 
the result of researches carried out quite recently. A few topics which 
are even rather speculative have been dealt with at some length 
(the theory of the motion of fronts in dynamic meteorology, given 
in Chapter 10.12, for example); others (like the theory of waves on 
sloping beaches) have been treated at some length as much because 
the author had a special fondness for the material as for their intrinsic 
mathematical interest. Thus the author has written a book which is 
rather personal in character, and which contains a selection of 
material chosen, very often, simply because it interested him, and 
he has allowed his predilections and tastes free rein. In addition, 
the book has a personal flavor from still another point of view since 
a quite large proportion of the material presented is based on the work 
of individual members of the Institute of Mathematical Sciences of 
New York University, and on theses and reports written by students 
attending the Institute. No attempt at completeness in citing the 
literature, even the more recent literature, was made by the author; 
on the other hand, a glance at the Bibliography (which includes 
only works actually cited in the book) will indicate that the recent 
literature has not by any means been neglected. 

In early youth by good luck the author came upon the writings 
of scientists of the British school of the latter half of the nineteenth 
century. The works of Tyndall, Huxley, and Darwin, in particular, 
made a lasting impression on him. This could happen, of course, only 
because the books were written in an understandable way and also 
in such a way as to create interest and enthusiasm: — but this was 
one of the principal objects of this school of British scientists. 
Naturally it is easier to write books on biological subjects for non- 
specialists than it is to write them on subjects concerned with the 
mathematical sciences — just because the time and effort needed to 
acquire a knowledge of modern mathematical tools is very great. 
That the task is not entirely hopeless, however, is indicated by John 
Tyndall's book on sound, which should be regarded as a great classic 
of scientific exposition. On the whole, the British school of popularizers 
of science wrote for people presumed to have little or no foreknow- 
ledge of the subjects treated. Now-a-days there exists a quite large 
potential audience for books on subjects requiring some knowledge 
of mathematics and physics, since a large number of specialists of 
all kinds must have a basic training in these disciplines. The author 
hopes that this book, which deals with so many phenomena of every 



XIV INTRODUCTION 

day occurrence in nature, might perhaps be found interesting, and 
understandable in some parts at least, by readers who have some 
mathematical training but lack specific knowledge of hydro- 
dynamics.* For example, the introductory discussion of waves on 
sloping beaches in Chapter 5, the purely geometrical discussion of 
the wave patterns created by moving ships in Chapter 8, great parts 
of Chapters 10 and 11 on waves in shallow water and flood waves in 
rivers, as well as the general discussion in Chapter 10 concerning 
the motion of fronts in the atmosphere, are in this category. 

2. Outline of contents 

It has already been stated that this book is planned as a coherent 
and unified whole in spite of the variety and diversity of its contents 
on both the mathematical and the physical sides. The possibility of 
achieving such a purpose lies in the fortunate fact that the material 
can be classified rather readily in terms of the types of mathematical 
problems which occur, and this classification also leads to a reasonably 
consistent ordering of the material with respect to the various types 
of physical problems. The book is divided into four main parts. 

Part I begins with a brief, but it is hoped adequate, development 
of the hydrodynamics of perfect incompressible fluids in irrotational 
flow without viscosity, with emphasis on those aspects of the subject 
relevant to flows with a free surface. Unfortunately, the basic general 
theory is unmanageable for the most part as a basis for the solution 
of concrete problems because the nonlinear free surface conditions 
make for insurmountable difficulties from the mathematical point 
of view. It is therefore necessary to make restrictive assumptions 
which have the effect of yielding more tractable mathematical 
formulations. Fortunately there are at least two possibilities in this 
respect which are not so restrictive as to limit too drastically the 
physical interest, while at the same time they are such as to lead to 
mathematical problems about which a great deal of knowledge is 
available. 

One of the two approximate theories results from the assumption 
that the wave amplitudes are small, the other from the assumption 

* The book by Rachel Carson [C.16] should be referred to here. This book is 
entirely nonmathematical, but it is highly recommended for supplementary 
reading. Parts of it are particularly relevant to some of the material in 
Chapter 6 of the present book. 



INTRODUCTION XV 

that it is the depth of the liquid which is small — in both cases, of 
course, the relevant quantities are supposed small in relation to some 
other significant length, such as a wave length, for example. Both of 
these approximate theories are derived as the lowest order terms 
of formal developments with respect to an appropriate small dimen- 
sionless parameter; by proceeding in this way, however, it can be 
seen how the approximations could be carried out to include higher 
order terms. The remainder of the book is largely devoted to the 
working out of consequences of these two theories, based on concrete 
physical problems: Part II is based on the small amplitude theory, 
and Part III deals with applications of the shallow water theory. 
In addition, there is a final chapter (Chapter 12) which makes up 
Part IV, in which a few problems are solved in terms of the basic 
general theory and the nonlinear boundary conditions are satisfied 
exactly; this includes a proof along lines due to Levi-Civita, of the 
existence, from the rigorous mathematical point of view, of progressing 
waves of finite amplitude. 

Part II, which is concerned with the first of the possibilities, 
might be called the linearized exact theory, since it can be obtained 
from the basic exact theory simply by linearizing the free surface 
conditions on the assumption that the wave motions studied con- 
stitute a small deviation from a constant flow with a horizontal free 
surface. Since we deal only with irrotational flows, the result is a 
theory based on the determination of a velocity potential in the space 
variables (containing the time as a parameter, however) as a solution 
of the Laplace equation satisfying certain linear boundary and initial 
conditions. This linear theory thus belongs, generally speaking, to 
potential theory. 

There is such a variety of material to be treated in Part II, which 
comprises Chapters 3 to 9, that a further division of it into sub- 
divisions is useful, as follows: 1) subdivision A, dealing with wave 
motions that are simple harmonic oscillations in the time; 2) sub- 
division B, dealing with unsteady, or transient, motions that arise 
from initial disturbances starting from rest; and 3) subdivision C, 
dealing with waves created in various ways on a running stream, 
in contrast with subdivisions A and B in which all motions are 
assumed to be small oscillations near the rest position of equilibrium 
of the fluid. 

Subdivision A is made up of Chapters 3, 4, and 5. In Chapter 3 
the basically important standing and progressing waves in liquids 



XVI INTRODUCTION 

of uniform depth and infinite lateral extent are treated; the important 
fact that these waves are subject to dispersion comes to light, and 
the notion of group velocity thus arises. The problem of the uniqueness 
of the solutions is considered — in fact, uniqueness questions are 
intentionally stressed throughout Part II because they are interesting 
mathematically and because they have been neglected for the most 
part until rather recently. It might seem strange that there could be 
any interesting unresolved uniqueness questions left in potential 
theory at this late date; the reason for it is that the boundary con- 
dition at a free surface is of the mixed type, i.e. it involves a linear 
combination of the potential function and its normal derivative, and 
this combination is such as to lead to the occurrence of non-trivial 
solutions of the homogeneous problems in cases which would in the 
more conventional problems of potential theory possess only iden- 
tically constant solutions. In fact, it is this mixed boundary con- 
dition at a free surface which makes Part II a highly interesting 
chapter in potential theory — quite apart from the interest of the 
problems on the physical side. Chapter 4 goes on to treat certain 
simple harmonic forced oscillations, in contrast with the free oscil- 
lations treated in Chapter 3. Chapter 5 is a long chapter which deals 
with simple harmonic waves in cases in which the depth of the water 
is not constant. A large part of the chapter concerns the propagation 
of progressing waves over a uniformly sloping beach; various methods 
of treating the problem are explained — in part with the object of 
illustrating recently developed techniques useful for solving boundary 
problems (both for harmonic functions and functions satisfying the 
reduced wave equation) in which mixed boundary conditions occur. 
Another problem treated (in Chapter 5.5) is the diffraction of waves 
around a vertical wedge. This leads to a problem identical with the 
classical diffraction problem first solved by Sommerfeld [S.12] for 
the special case of a rigid half-plane barrier. Here again the uniqueness 
question comes to the fore, and, as in many of the problems of Part II, 
it involves consideration of so-called radiation conditions at infinity. A 
uniqueness theorem is derived and also a new, and quite simple and 
elementary, solution for Sommerfeld's diffraction problem is given. 
It is a curious fact that these gravity wave problems, the solutions 
of which are given in terms of functions satisfying the Laplace 
equation, nevertheless require for the uniqueness of the solutions 
that conditions at infinity of the radiation type, just as in the more 
familiar problems based on the linear wave equation, be imposed; 



INTRODUCTION XVII 

ordinarily in potential theory it is sufficient to require only boundednes s 
conditions at infinity to ensure uniqueness. 

In subdivision B of Part II, comprised of Chapter 6, a variety of 
problems involving transient motions is treated. Here initial con- 
ditions at the time t = are imposed. The technique of the Fourier 
transform is explained and used to obtain solutions in the form of 
integral representations. The important classical cases (treated first 
by Cauchy and Poisson) of the circular waves due to disturbances at 
a point of the free surface in an infinite ocean are studied in detail. 
For this purpose it is very useful to discuss the integral representations 
by using an asymptotic approximation due to Kelvin (and, indeed, 
developed by him for the purpose of discussing the solutions of just 
such surface wave problems) and called the principle, or method, of 
stationary phase. These results then can be interpreted in a striking 
way in terms of the notion of group velocity. Recently there have 
been important applications of these results in oceanography: one 
of them concerns the type of waves called tsunamis, which are 
destructive waves in the ocean caused by earthquakes, another 
concerns the location of storms at sea by analyzing wave records 
on shore in the light of the theory at present under discussion. The 
question of uniqueness of the transient solutions — again a problem 
solved only recently — is treated in the final section of Chapter 6. 
An opportunity is also afforded for a discussion of radiation con- 
ditions (for simple harmonic waves) as limits as t -> oo in appropriate 
problems concerning transients, in which boundedness conditions at 
infinity suffice to ensure uniqueness. 

The final subdivision of Part II, subdivision C, deals with small 
disturbances created in a stream flowing initially with uniform 
velocity and with a horizontal free surface. Chapter 7 treats waves in 
streams having a uniform depth. Again, in the case of steady motions, 
the question of appropriate conditions of the radiation type arises; 
the matter is made especially interesting here because the circum- 
stances with respect to radiation conditions depend radically on the 
parameter U 2 jgh, with U and h the velocity and depth at infinity, res- 
pectively. Thus if U 2 /gh > 1, no radiation conditions need be im- 
posed, if U 2 /gh < 1 they are needed, while if U 2 /gh = 1 something 
quite exceptional occurs. These matters are studied, and their physical 
interpretations are discussed in Chapter 7.3 and 7.4. In Chapter 8 
Kelvin's theory of ship waves for the idealized case of a ship regarded 
as a point disturbance moving over the surface of the water is treated 



XVIII INTRODUCTION 

in considerable detail. The principle of stationary phase leads to a 
beautiful and elegant treatment of the nature of ship waves that is 
purely geometrical in character. The cases of curved as well as 
straight courses are considered, and photographs of ship waves taken 
from airplanes are reproduced to indicate the good accord with 
observations. Finally, in Chapter 9 a general theory (once more the 
result of quite recent investigations ) for the motion of ships, regarded 
as floating rigid bodies, is presented. In this theory no restrictive 
assumptions — regarding, for example, the coupling (or lack of 
coupling, as in an old theory due to Krylov [K.20] between the 
motion of the sea and the motion of the ship, or between the various 
degrees of freedom of the ship — are made other than those needed to 
linearize the problem. This means essentially that the ship must be 
regarded as a thin disk so that it can slice its way through the water 
(or glide over the surface, perhaps) with a finite velocity and still 
create waves which do not have large amplitudes; in addition, it 
is necessary to suppose that the motion of the ship is a small oscil- 
lation relative to a motion of translation with uniform velocity. The 
theory is obtained by making a formal development of all conditions 
of the complete nonlinear boundary problem with respect to a para- 
meter which is a thickness-length ratio of the ship. The resulting 
theory contains the classical Michell-Havelock theory for the wave 
resistance of a ship in terms of the shape of its hull as the simplest 
special case. 

We turn next to Part III, which deals with applications of the 
approximate theory which results from the assumption that it is the 
depth of the liquid which is small, rather than the amplitude of the 
surface waves as in Part II. The theory, called here the shallow 
water theory, leads to a system of nonlinear partial differential 
equations which are analogous to the differential equations for the 
motion of compressible gases in certain cases. We proceed to outline 
the contents of Part III, which is composed of two long chapters. 

In Chapter 10 the mathematical methods based on the theory of 
characteristics are developed in detail since they furnish the basis 
for the discussion of practically all problems in Part III; it is hoped 
that this preparatory discussion of the mathematical tools will make 
Part III of the book accessible to engineers and others who have not 
had advanced training in mathematical analysis and in the methods 
of mathematical physics. In preparing this part of the book the 
author's task was made relatively easy because of the existence of the 



INTRODUCTION XIX 

book by Courant and Friedrichs [C.9], which deals with gas dynamics; 
the presentation of the basic theory given here is largely modeled 
on the presentation given in that book. The concrete problems dealt 
with in Chapter 10 are quite varied in character, including the 
propagation of disturbances into still water, conditions for the 
occurrence of a bore and a hydraulic jump (phenomena analogous to 
the occurrence of shock waves in gas dynamics), the motion resulting 
from the breaking of a dam, steady two dimensional motions at 
supercritical velocity, and the breaking of waves in shallow water. 
The famous problem of the solitary wave is discussed along the lines 
used recently by Friedrichs and Hyers [F.13] to prove rigorously 
the existence of the solitary wave from the mathematical point of 
view; this problem requires carrying the perturbation series which 
formulate the shallow water theory to terms of higher order. The 
problem of the motion of frontal discontinuities in the atmosphere, 
which lead to the development of cyclonic disturbances in middle 
latitudes, is given a formulation — on the basis of hypotheses which 
simplify the physical situation — which brings it within the scope 
of a more general "shallow water theory". Admittedly (as has already 
been noted earlier) this theory is somewhat speculative, but it is 
nevertheless believed to have potentialities for clarifying some of 
the mysteries concerning the dynamical causes for the development 
and deepening of frontal disturbances in the atmosphere, especially 
if modern high speed digital computing machines are used as an aid 
in solving concrete problems numerically. 

Chapter 10 concludes with the discussion of a few applications of 
the linearized version of the shallow water theory. Such a linearization 
results from assuming that the amplitude of the waves is small. The 
most famous application of this theory is to the tides in the oceans 
(and also in the atmosphere, for that matter); strange though it 
seems at first sight, the oceans can be treated as shallow for this 
phenomenon since the wave lengths of the motions are very long 
because of the large periods of the disturbances caused by the moon 
and the sun. This theory, as applied to the tides, is dealt with only 
very summarily, since an extended treatment is given by Lamb 
[L.3]. Instead, some problems connected with the design of floating 
breakwaters in shallow water are discussed, together with brief 
treatments of the oscillations in certain lakes (the lake at Geneva 
in Switzerland, for example) called seiches, and oscillations in harbors. 

Finally, Part III concludes with Chapter 11 on the subject of 



XX INTRODUCTION 

mathematical hydraulics, which is to be understood here as referring 
to flows and wave motions in rivers and other open channels with 
rough sides. The problems of this chapter are not essentially different, 
as far as mathematical formulations go, from the problems treated 
in the preceding Chapter 10. They differ, however, on the physical 
side because of the inclusion of a force which is just as important as 
gravity, namely a force of resistance caused by the rough sides and 
bottom of the channels. This force is dealt with empirically by 
adding a term to the equation expressing the law of conservation of 
momentum that is proportional to the square of the velocity and 
with a coefficient depending on the roughness and the so-called 
hydraulic radius of the channel. The differential equations remain of 
the same type as those dealt with in Chapter 10, and the same under- 
lying theory based on the notion of the characteristics applies. 

Steady motions in inclined channels are first dealt with. In par- 
ticular, a method of solving the problem of the occurrence of roll 
waves in steep channels is given; this is done by constructing a 
progressing wave by piecing together continuous solutions through 
bores spaced at periodic intervals. This is followed by the solution 
of a problem of steady motion which is typical for the propagation 
of a flood down a long river; in fact, data were chosen in such a way 
as to approximate the case of a flood in the Ohio River. A treatment 
is next given for a flood problem so formulated as to correspond 
approximately to the case of a flood wave moving down the Ohio 
to its junction with the Mississippi, and with the result that distur- 
bances are propagated both upstream and downstream in the Missis- 
sippi and a backwater effect is noticeable up the Ohio. In these 
problems it is necessary to solve the differential equations numerically 
(in contrast with most of the problems treated in Chapter 10, in 
which interesting explicit solutions could be given), and methods of 
doing so are explained in detail. In fact, a part of the elements of 
numerical analysis as applied to solving hyperbolic partial differential 
equations by the method of finite differences is developed. The results 
of a numerical prediction of a flood over a stretch of 400 miles in 
the Ohio River as it actually exists are given. The flood in question 
was the 1945 flood — one of the largest on record — and the predic- 
tions made (starting with the initial state of the river and using the 
known flows into it from tributaries and local drainage) by numerical 
integration on a high speed digital computer (the Univac) check 
quite closely with the actually observed flood. Numerical predictions 



INTRODUCTION XXI 

were also made for the case of a flood (the 1947 flood in this case) 
coming down the Ohio and passing through its junction with the 
Mississippi; the accuracy of the prediction was good. This is a case 
in which the simplified methods of the civil engineers do not work 
well. These results, of course, have important implications for the 
practical applications. 

Finally Part IV, made up of Chapter 12, closes the book with a 
few solutions based on the exact nonlinear theory. One class of problems 
is solved by assuming a solution in the form of power series in the 
time, which implies that initial motions and motions for a short time 
only can be determined in general. Nevertheless, some interesting 
cases can be dealt with, even rather easily, by using the so-called 
Lagrange representation, rather than the Euler representation which 
is used otherwise throughout the book. The problem of the breaking 
of a dam. and, more generally, problems of the collapse of columns 
of a liquid resting on a rigid horizontal plane can be treated in this 
way. The book ends with an exposition of the theory due to Levi- 
Civita concerning the problem of the existence of progressing waves of 
finite amplitude in water of infinite depth which satisfy exactly the 
nonlinear free surface conditions. 






Acknowledgments 



Without the support of the Mathematics Branch and the Mechanics 
Branch of the Office of Naval Research this book would not have been 
written. The author takes pleasure in acknowledging the help and 
encouragement given to him by the ONR in general, and by Dr. Joa- 
chim Weyl, Dr. Arthur Grad, and Dr. Philip Eisenberg in particular. 
Although she is no longer working in the ONR, it is neverthe- 
less appropriate at this place to express special thanks to Dean 
Mina Rees, who was head of the Mathematics Branch when this 
book was begun. 

Among those who collaborated with the author in the preparation 
of the manuscript, Dr. Andreas Troesch should be singled out for 
special thanks. His careful and critical reading of the manuscript re- 
sulted in many improvements and the uncovering and correction of 
errors and obscurities of all kinds. Another colleague, Professor E. 
Isaacson, gave almost as freely of his time and attention, and also 
aided materially in revising some of the more intricate portions of the 
book. To these fellow workers the author feels deeply indebted. 

Miss Helen Samoraj typed the entire manuscript in a most efficient 
(and also good-humored) way, and uncovered many slips and in- 
consistencies in the process. 

The drawings for the book were made by Mrs. Beulah Marx and 
Miss Larkin Joyner. The index was prepared by Dr. George Booth and 
Dr. Walter Littman with the assistance of Mrs. Halina Montvila. 

A considerable part of the material in the present book is the result 
of researches carried out at the Institute of Mathematical Sciences of 
New York University as part of its work under contracts with the 
Office of Naval Research of the U.S. Department of Defense, and to a 
lesser extent under a contract with the Ohio River Division of the 
Corps of Engineers of the U.S. Army. The author wishes to express his 
thanks generally to the Institute; the cooperative and friendly spirit 
of its members, and the stimulating atmosphere it has provided have 
resulted in the carrying out of quite a large number of researches in 
the field of water waves. A good deal of these researches and new 
results have come about through the efforts of Professors K. O. Fried- 



XXIV ACKNOWLEDGMENTS 

richs, Fritz John, J. B. Keller, H. Lewy (of the University of Cali- 
fornia), and A. S. Peters, together with their students or with visitors 
at the Institute. 

J. J. Stoker 
New York, N.Y. 
January, 1957. 



Contents 



PART I 

CHAPTER PAGE 

Introduction ix 

Acknowledgments xxiii 

1. Basic Hydrodynamics 3 

1.1 The laws of conservation of momentum and mass 3 

1.2 Helmholtz's theorem 7 

1.3 Potential flow and Bernoulli's law 9 

1.4 Boundary conditions 10 

1.5 Singularities of the velocity potential 12 

1.6 Notions concerning energy and energy flux 13 

1.7 Formulation of a surface wave problem 15 

2. The Two Basic Approximate Theories 19 

2.1 Theory of waves of small amplitude 19 

2.2 Shallow water theory to lowest order. Tidal theory .... 22 

2.3 Gas dynamics analogy 25 

2.4 Systematic derivation of the shallow water theory 27 



PART II 

Subdivision A 
Waves Simple Harmonic in the Time 

3. Simple Harmonic Oscillations in Water of Constant Depth . . 37 

3.1 Standing waves 37 

3.2 Simple harmonic progressing waves 45 

3.3 Energy transmission for simple harmonic waves of small ampli- 
tude 47 

3.4 Group velocity. Dispersion 51 

4. Waves Maintained by Simple Harmonic Surface Pressure in 

Water of Uniform Depth. Forced Oscillations 55 

4.1 Introduction 55 

4.2 The surface pressure is periodic for all values of a? 57 

XXV 



XXVI CONTENTS 

CHAPTER PAGE 

4.3 The variable surface pressure is confined to a segment of the 
surface 58 

4.4 Periodic progressing waves against a vertical cliff 67 

5. Waves on Sloping Beaches and Past Obstacles 69 

5.1 Introduction and summary 69 

5.2 Two-dimensional waves over beaches sloping at angles a> = n/2n 77 

5.3 Three-dimensional waves against a vertical cliff 84 

5.4 Waves on sloping beaches. General case 95 

5.5 Diffraction of waves around a vertical wedge. Sommerfeld's 
diffraction problem 109 

5.6 Brief discussions of additional applications and of other methods 

of solution 133 



Subdivision B 
Motions Starting from Rest. Transients 



6. Unsteady Motions 



149 



6.1 General formulation of the problem of unsteady motions . . 149 

6.2 Uniqueness of the unsteady motions in bounded domains . . 150 

6.3 Outline of the Fourier transform technique 153 

6.4 Motions due to disturbances originating at the surface ... 156 

6.5 Application of Kelvin's method of stationary phase .... 163 

6.6 Discussion of the motion of the free surface due to disturbances 
initiated when the water is at rest 167 

6.7 Waves due to a periodic impulse applied to the water when 
initially at rest. Derivation of the radiation condition for purely 
periodic waves 174 

6.8 Justification of the method of stationary phase 181 

6.9 A time-dependent Green's function. Uniqueness of unsteady 
motions in unbounded domains when obstacles are present . 187 



Subdivision C 
Waves on a Running Stream. Ship Waves 

7. Two-dimensional Waves on a Running Stream in Water of 

Uniform Depth 198 

7.1 Steady motions in water of infinite depth with p — on the 

free surface 199 



CONTENTS XXVII 

CHAPTER PAGE 

7.2 Steady motions in water of infinite depth with a disturbing pres- 
sure on the free surface 201 

7.3 Steady waves in water of constant finite depth 207 

7.4 Unsteady waves created by a disturbance on the surface of a 
running stream 210 

8. Waves Caused by a Moving Pressure Point. Kelvin's Theory of 

the Wave Pattern created by a Moving Ship 219 

8.1 An idealized version of the ship wave problem. Treatment by 

the method of stationary phase 219 

8.2 The classical ship wave problem. Details of the solution . . 224 

9. The Motion of a Ship, as a Floating Rigid Body, in a Seaway 245 

9.1 Introduction and summary 245 

9.2 General formulation of the problem 264 

9.3 Linearization by a formal perturbation procedure 269 

9.4 Method of solution of the problem of pitching and heaving of a 
ship in a seaway having normal incidence 278 



PART III 

10. Long Waves in Shallow Water 291 

10.1 Introductory remarks and recapitulation of the basic equations 291 

10.2 Integration of the differential equations by the method of char- 
acteristics 293 

10.3 The notion of a simple wave 300 

10.4 Propagation of disturbances into still water of constant depth 305 

10.5 Propagation of depression waves into still water of constant 
depth 308 

10.6 Discontinuity, or shock, conditions 314 

10.7 Constant shocks: bore, hydraulic jump, reflection from a rigid 
wall 326 

10.8 The breaking of a dam 333 

10.9 The solitary wave 342 

10.10 The breaking of waves in shallow water. Development of bores 351 

10.11 Gravity waves in the atmosphere. Simplified version of the 
problem of the motion of cold and warm fronts 374 

10.12 Supercritical steady flows in two dimensions. Flow around 
bends. Aerodynamic applications 405 

10.13 Linear shallow water theory. Tides. Seiches. Oscillations in 
harbors. Floating breakwaters 414 



XXVIII CONTENTS 

CHAPTER PAGE 

11. Mathematical Hydraulics 451 

11.1 Differential equations of flow in open channels 452 

11.2 Steady flows. A junction problem 456 

11.3 Progressing waves of fixed shape. Roll waves 461 

11.4 Unsteady flows in open channels. The method of characteristics 469 

11.5 Numerical methods for calculating solutions of the differential 
equations for flow in open channels 474 

11.6 Flood prediction in rivers. Floods in models of the Ohio River 

and its junction with the Mississippi River 482 

11.7 Numerical prediction of an actual flood in the Ohio, and at its 
junction with the Mississippi. Comparison of the predicted with 

the observed floods 498 

Appendix to Chapter 11. Expansion in the neighborhood of the first 

characteristic 505 



PART IV 

12. Problems in which Free Surface Conditions are Satisfied Exactly. 

The Breaking of a Dam. Levi-Civita's Theory 513 

12.1 Motion of water due to breaking of a dam, and related problems 513 

12.2 The existence of periodic waves of finite amplitude .... 522 

12.2a Formulation of the problem 522 

12.2b Outline of the procedure to be followed in proving the existence 

of the function co(%) 526 

12.2c The solution of a class of linear problems 529 

12. 2d The solution of the nonlinear boundary value problem . . . 537 

Bibliography 545 

Author Index 561 

Subject Index 563 



PART I 



CHAPTER 1 



Basic Hydrodynamics 

1.1. The laws of conservation of momentum and mass 

As has been stated in the introduction, we deal exclusively in this 
book with flows in water (and air) which are of such a nature as 
to make it unnecessary to take into account the effects of viscosity 
and compressibility. As a consequence of the neglect of internal 
friction, or in other words of neglect of shear stresses, it is well 
known that the stress system* in the liquid is a state of uniform 
compression at each point. The intensity of the compressive stress 
is called the pressure p. 

The equation of motion of a fluid particle can then be obtained on 
the basis of Newton's law of conservation of momentum, as follows. 
A small rectangular element of the fluid is shown in Figure 1.1.1 



P + Pv8x 



Sz 



->- x 



Fig. 1.1.1. Pressure on a fluid element 

with the pressure acting on the faces normal to the #-axis. Newton's 
law for the ^-direction is then 

[ — (P + Vx <5«) + V]fy $ z + Xq dec 6y Sz — ga {x) dx dy Sz 



* We assume that the usual concepts of the general mechanics of continuous 
media are known. 



WATER WAVES 



in which X is the external or body force component per unit mass 
and a (x) is the acceleration component, both in the ^-direction, and 
q is the density. The quantities p, X, and a (x) are in general functions 
of x, y, z, and t. Here, as always, we shall use letter subscripts to 
denote differentiation, and this accounts for the symbol a (x) to denote 
the component of a vector in the ^-direction. Upon passing to the 
limit in allowing dx, dy, dz to approach zero we obtain the equation 
of motion for the ^-direction in the form — p x -f gX = ga (x) , and 
analogous expressions for the two other directions. Thus we have the 
equations of motion 



(l.i.i) 

or, in vector form: 
(1.1.2) 



~Vx + X = a {x) , 

1 
— Vv + Y = *(,). 



— Vz +Z = a {z) , 



1 



grad p + F = a, 



with an obvious notation. The body force F plays a very important 
role in our particular branch of hydrodynamics — in fact the main 
results of the theory are entirely conditioned by the presence of the 
gravitational force F = (0, — g, 0), in which g represents the acceler- 
ation of gravity. It should be observed that we consider the positive 
y-axis to be vertically upward, and the x, z-plane therefore to be horizontal 
(usually it will be taken as the undisturbed water surface). This con- 
vention regarding the disposition of the coordinate axes will be main- 
tained, for the most part, throughout the book. 

The differential equations (1.1.1) are in what is called the Lagrang- 
ian form, in which one has in mind a direct description of the motion 
of each individual fluid particle as a function of the time. It is more 
useful for most purposes to work with the equations of motion in the 
so-called Eulerian form. In this form of the equations one concen- 
trates attention on the determination of the velocity distribution in 
the region occupied by the fluid without trying to follow the motion of 
the individual fluid particles, but rather observing the velocity 
distribution at fixed points in space as a function of the time. In 



BASIC HYDRODYNAMICS 5 

other words, the velocity field, with components u, v, w, is to be 
determined as a function of the space variables and the time. After- 
wards, if that is desired, the motion of the individual particles can 
be obtained by integrating the system of ordinary differential equa- 
tions x = u, y = v, z = w, in which the dot over the quantities x, y, 
z means differentiation with respect to the time in following the 
motion of an individual particle. 

In order to restate the equations of motion (1.1.1) in terms of the 
Euler variables u, v, w, and in order to carry out other important 
operations as well, it is necessary to calculate time derivatives of 
various functions associated with a given fluid particle in following 
the motion of the particle. For example, we need to calculate the 
time derivative of the velocity of a particle in order to obtain the 
acceleration components occurring in (1.1.1), and quite a few other 
quantities will occur later on for which such particle derivatives will 
be needed. Suppose, then, that F(x, y,z;t) is a function associated 
with a particle which follows the path given by the vector 

x= (x(t), y(t),z(t)); 
it follows that 

x = (x{t), y{t), z(t)) = (u, v, w) 

is the velocity vector associated with the particle. For this particle 
the arguments x, y, z of the function F are of course the functions 
of t which characterize the motion of the particle; as a consequence 
we have 

dF 

— = F x x + F y y + F z z + F t 
at 

= uF x +vF y +wF z +F t , 

and hence the operation of taking the particle derivative djdt is 
defined as follows: 

(1.1-3) j f ( ) = u( ).+*( ) y +w( ). + ( ) f . 

The distinction between dF/dt and dFjdt = F t should be carefully 
noted. 

Since the acceleration a of a particle is given by a = (du/dt, dv/dt, 
dw/dt), in which (u, v, w) are the components of the velocity v of 



WATER WAVES 



the particle, it follows from (1.1.3) that the component a {x) = du/dt 
is given by 

du 

— = uu x + vu y + wu z + u t , 
at 

with similar expressions for the other components. The equations of 
motion (1.1.1) are therefore given as follows in terms of the Euler 
variables : 



(1.1.4; 



when we specify the external or body force to consist only of the 
force of gravity. 

Equations (1.1.4) form a set of three nonlinear partial differential 
equations for the five quantities u, v, w, q, and p. Since the fluid is 
assumed to be incompressible, the density q can be taken as a known 
constant. At the same time, the assumption of incompressibility leads 
to a relatively simple differential equation expressing the law of 
conservation of mass, and this equation constitutes the needed fourth 
equation for the determination of the velocity components and the 
pressure. Perhaps the simplest way to derive the mass conservation 
law is to start from the relation 



u t 


+ 


uu x 


+ 


VUy 


+ 


wu z 


= 


1 

Q 


Vx 


v t 


+ 


uv x 


+ 


vv y 


+ 


wv z 


= 


1 

Q 


Vv 


w t 


+ 


uw a 


+ 


vw % 


+ 


ww z 


= 


1 

Q 


Vz 



/J 



QV n 



dS = 0, 



which states that the mass flux outward through any fixed closed 
surface enclosing a region in which no liquid is created or destroyed 
is zero. (By v n we mean the velocity component taken positive in the 
direction of the outward normal to the surface.) An application of 
Gauss's divergence theorem: 

(1.1.5) (Lv n dS = JJJdiv (qv) dx 

S R 

to the above integral leads to the relation 

fffdiv (qv) dx = 

R 



BASIC HYDRODYNAMICS 7 

for any arbitrary region R. It follows therefore that div (qv) — 
everywhere, and since g = constant, we have finally 

(1.1.6) div v = u x + v y + w z = 

as the expression of the law of conservation of mass. The equation 
(1.1.6) is also frequently called the equation of continuity. 

Equations (1.1.4) and (1.1.6) are sufficient, once appropriate 
initial and boundary conditions (to be discussed shortly) are imposed, 
to determine the velocity components u, v, w, and the pressure p 
uniquely. 



1.2. Helmholtz's theorem 

Before discussing boundary conditions it is preferable to for- 
mulate a few additional conservation laws which are consequences 
of the assumptions made so far— in particular of the assumption that 
internal fluid friction can be neglected. 

The first of these laws to be discussed is the law of conservation 
of circulation. The notion of circulation is defined as follows. Consider 
a closed curve C which moves with the fluid (that is, C consists 
always of the same particles of the fluid). The circulation r = r(t) 
around C is defined by the line integral 



1.2.1) r(t) = Judx +vdy + w dz 

c 



<p v s ds 



in which v s is the velocity component of the fluid tangent to C, 

and ds is the element of arc length of C. The curve C is considered 

as given by the vector x(or, t) with a a parameter on C such that 

5^ a ^ 1 and x(0, t) = x(l, t). We are thus operating in terms of 

the Lagrange system of variables rather than in terms of the Euler 

system, and fixing a value of a has the effect of picking out a specific 

particle on C. 

l 

We may write r(t) = v • x a da in which v • x a is a scalar product 
o 
and x CT , as usual, refers to differentiation with respect to a. For the 

time derivative P we have therefore 



WATER WAVES 



f(t) = /( 



v • X, 



v • x a )da. 



From the equation of motion (1.1.2) in the Lagrangian form with 
a = v, F = (0, — g, 0) = — grad (gy), and from x a = v a , the last 
equation yields 

l 

- x a • grad p — gx a • grad y + v • v CT do 
Q 



(i.2.2) r(t) 



i. 

\ 






- Vo - gy a + - (v • v) ff 
e 2 



do* 



since the values of p, y, and v coincide at g = and cr = 1, and q 
and g are constants. The last equation evidently states that in a 
nonviscous fluid the circulation around any closed curve consisting of 
the same fluid particles is constant in time. This is the theorem of Helm- 
holtz. The assumption of zero viscosity entered into our derivation 
through the use of (1.1.2) as equation of motion.* 

In this book we are interested in the special case in which the 
circulation for all closed curves is zero. This case is very important 
in the applications because it occurs whenever the fluid is assumed 
to have been at rest or to have been moving with a constant velocity 
at some particular time, so that v = const, holds at that time, and 
hence r vanishes for all time. The cases in which the fluid motion 
begins from such states are obviously very important. 

The assumption that T vanishes for all closed curves has a number 
of consequences which are basic for all that follows in this book. 
The first conclusion from T = follows almost immediately from 
Stokes's theorem: 



(1.2.3; 



v Q ds 



//(curl 



dA. 



in which the surface integral is taken over any surface S spanning the 
curve C. If r = for all curves C, as we assume, it follows easily by a 
well-known argument that the vector curl v vanishes everywhere: 

* It should be added that the law of conservation of circulation holds under 
much wider conditions than were assumed here (cf. [C.9], p. 19). 



BASIC HYDRODYNAMICS 9 

(1.2.4) curl v = (w y — v z , u z —w x , v x —u y ) = 0, 

and the flow is then said to be irrotational. In other words, a motion 
in a nonviscous fluid which is irrotational at one instant always 
remains irrotational. Throughout the rest of this hook we shall assume 
all flows to he irrotational. 

1.3. Potential flow and Bernoulli's law 

The assumption of irrotational flow results in a number of sim- 
plifications in our theory which are of the greatest utility. In the 
first place, the fact that curl v = (cf. (1.2.4)) ensures, as is well 
known, the existence of a single- valued velocity potential 0(cc, y, z; t) 
in any simply connected region, from which the velocity field can be 
derived by taking the gradient: 

(1.3.1) v = grad0 = (0 X , y , Z ), 
or, in terms of the components of v: 

(1.3.2) u = X , v = y , w = Z . 

The velocity potential is, indeed, given by the line integral 

x, y,z 

0(cc, y, z; t) = I u dx + v dy -\- w dz. 

The vanishing of curl v ensures that the expression to be integrated 
is an exact differential. Once it is known that the velocity com- 
ponents are determined by (1.3.2), it follows from the continuity 
equation (1.1.6), i.e. div v = 0, that the velocity potential is a 
solution of the Laplace equation 

(1.3.3) V 2 = XX + yy + ZZ = 0, 

as one readily sees, and is thus a harmonic function. This fact 
represents a great simplification, since the velocity field is derivable 
from a single function satisfying a linear differential equation which 
has been very much studied and about which a great deal is known. 
Still another important consequence of the irrotational character 
of a flow can be obtained from the equations of motion (1.1.4). By 
making use of (1.2.4), it is readily verified that the equations of 
motion (1.1.4) can be written in the following vector form: 

1 p 

grad t + - grad (u 2 + v 2 + w 2 ) = — grad - — grad (gy). 

2 ' Q 

use having been made of the fact that g = constant. Integration 



10 WATER WAVES 

of this relation leads to the important equation expressing what is 
called Bernoulli's law: 

(1.3.4) t + 1 (u* + v* + w*) + ? + gy = C(t), 

2 Q 

in which C(t) may depend on t, but not on the space variables. There 
are two other forms of Bernoulli's law for the case of steady flows, 
one of which applies along stream lines even though the flow is 
not irrotational, but since we make no use of these laws in this book 
we refrain from formulating them. 

The potential equation (1.3.3) together with Bernoulli's law 
(1.3.4) can be used to take the place of the equations of motion 
(1.1.4) and the continuity equation (1.1.6) as a means of determining 
the velocity components u, v, w, and the pressure p: in effect, u, v, 
and w are determined from the solution of (1.3.3), after which the 
pressure p can be obtained from (1.3.4). It is true that the pressure 
appears to be determined only within a function which is the same 
at each instant throughout the fluid. On physical grounds it is, 
however, clear that a function of t alone added to the pressure p 
has no effect on the motion of the fluid since no pressure gradients 

result from such an addition to the pressure. In fact, if we set 

t 

= 0* 4- \C(£) d£, then 0* is a harmonic function with 



+ JC(S) d£ 



grad = grad 0* and the Bernoulli law with reference to it has a 
vanishing right hand side. Thus we may take C(t) = in (1.3.4) 
without any essential loss of generality. 

While it is true that the Laplace equation is a linear differential 
equation, it does not follow that we shall be able to escape all of the 
difficulties arising from the nonlinear character of the basic differen- 
tial equations of motion (1.1.4). As we shall see, the problems of 
interest here remain essentially nonlinear because the Bernoulli law 
(1.3.4), and another condition to be derived in the next section, give 
rise to nonlinear boundary conditions at free surfaces. In the next 
section we take up the important question of the boundary con- 
ditions appropriate to various physical situations. 

1.4. Boundary conditions 

We assume the fluid under consideration to have a boundary 
surface S, fixed or moving, which separates it from some other 
medium, and which has the property that any particle which is once 



BASIC HYDRODYNAMICS 11 

on the surface remains on it.* Examples of such boundary surfaces 
of importance for us are those in which S is the surface of a fixed 
rigid body in contact with the fluid — the bottom of the sea, for 
example— or the free surface of the water in contact with the air. 
If such a surface S were given, for example, by an equation 
£(x, y, z; t) = 0, it follows from (1.1.3) that the condition 

(1.4.1) d f=u£ x + vC y + «£, + St = 

at 

would hold on S. From (1.3.2) and the fact that the vector (£ x , £ v , £ 2 ) 
is a normal vector to S it follows that the condition (1.4.1) can be 
written in the form 

B0 Ct 

(1.4.2) 



dn vci + c ; + a 

in which d/dn denotes differentiation in the direction of the normal 
to S and v n means the common velocity of fluid and boundary 
surface in the direction normal to the surface. 

In the important special case in which the boundary surface S 
is fixed, i.e. it is independent of the time t, we have the condition 

30 

(1.4.3) — = on S. 

on 

This is the appropriate boundary condition at the bottom of the sea, 
or at the walls of a tank containing water. 

Another extremely important special case is that in which S is a 
free surface of the liquid, i.e. a surface on which the pressure p is 
prescribed but the form of the surface is not prescribed a priori. 
We shall in general assume that such a free surface is given by the 
equation 

(1.4.4) y = 7}(x,z; t). 

On such a surface £ = y — rj(x, z; t) = for any particle, and hence 
(1.4.1) yields the condition 

(1.4.5) x rj x -& y + z rj z + rj t = on S. 

In addition, as remarked above, we assume that the pressure p is 
given on S; as a consequence the Bernoulli law (1.3.4) yields the 
condition: 

* Actually, this property is a consequence of the basic assumption in con- 
tinuum mechanics that the motion of the fluid can be described mathematically 
as a topological deformation which depends continuously on the time t. 



12 WATER WAVES 

(1.4.6) grj + 0,+1 (01 + 0\ + 01) + ? = on 5. 
2 £ 

(As remarked earlier, we may take the quantity C(t) = in (1.3.4).) 
Thus the potential function must satisfy the two nonlinear boundary 
conditions (1.4.5) and (1.4.6) on a free surface. This is in sharp con- 
trast to the single linear boundary condition (1.4.3) for a fixed 
boundary surface, but it is not strange that two conditions should 
be prescribed in the case of the free surface since an additional 
unknown function rj(x, z; t), the vertical displacement of the free 
surface, is involved in the latter case. 

Later on we shall also be concerned with problems involving rigid 
bodies floating in the water and S will be the portion of the rigid 
body in contact with the water. In such cases the function rj(x, z; t) 
will be determined by the motion of the rigid body, which in turn 
will be fixed (through the dynamical laws of rigid body mechanics) 
by the pressure p between the body and the water in accordance 
with (1.4.6). The detailed conditions for such cases will be worked 
out later on at an appropriate place. 

1.5. Singularities of the velocity potential 

In our discussion up to now it has been tacitly assumed that all 
quantities such as the pressure, velocity potential, velocity com- 
ponents, etc. are regular functions of their arguments. It is, however, 
often useful to permit singularities of one kind or another to occur 
as an idealization of, or an approximation to, certain physical situations. 
Perhaps the most useful such singularity is the point source or sink 
which is given by the harmonic function 

(1.5.1) 0=H£ t r 2 = x 2 -\-y 2 + z 2 

4>jir 

in three dimensions, and by 

(1.5.2) = — log r, r 2 = x 2 + y 2 

in two dimensions. Both of these functions yield flows which are 
radially outward from the origin, and for which the flux per unit 
time across a closed surface (for (1.5.1)) or a closed curve (for (1.5.2)) 
surrounding the origin has the value c, as one readily verifies since 
d0/dn = d0/dr for r = constant. That these functions represent at 



BASIC HYDRODYNAMICS 13 

best idealizations of the physical situations implied in the words 
source and sink is clear from the fact that they yield infinite velocities 
at r = 0. Nevertheless, it is very useful here — as in other branches 
of applied mathematics —to accept such infinities with the reservation 
that the results obtained are not to be taken too literally in the 
immediate vicinity of the singular point. 

We shall have occasion to deal with other singularities than sources 
or sinks, such as dipoles and multipoles, but these will be introduced 
when needed. 



1.6. Notions concerning energy and energy flux 

In dealing with surface gravity waves in water it is important 
and useful to analyze in some detail the flow of energy in the fluid 
past a given surface S. Let R be the region occupied by water and 
bounded by a "geometric" surface S which may, or may not, move 
independently of the liquid. The energy E contained in R consists 
of the kinetic energy of the water particles in R and their potential 
energy due to gravity; hence E is given by 



(1.6.1) E 



/// 



\m+4>l+<i>l)+gy 



dx dy dz, 



or, alternatively, by 

(1.6.2) E = - J*JJ(p + 9 t )dx dy dz 

R 

upon applying Bernoulli's law (1.3.4) with C(t) = 0. 

We wish to calculate dE/dt, having in mind that the region R 
is not necessarily fixed, but may depend on the time t. Quite generally, 

if E = I f(x, y, z; t)dx dy dz, it is well known that 

R 

~dt = \\\ U dX dy dZ + \\ fVn dS 

R S 

in which v n denotes the normal velocity of the boundary S of R 
taken positive in the direction outward from R. In applying the 
formula for dE/dt we make use of the definition of the function / 
implied in (1.6.1) in the first term, but take / from (1.6.2) for the 
second term. The result is 



14 WATER WAVES 



dE 
~dt 



- jj(p+Q0 t )v n dS. 



s 
The integrand in the first integral can be expressed in the form 

#.(**). + ® v (&t)y + &z(&t)z = g r ad • grad t 
and hence the integral can be written as the following surface integral: 

30 



- 



dS, 
on 



s 
in view of Green's formula and the fact that V 2 = 0. Thus the 
expression for dEjdt, the rate of change of the energy in R, can be 
put into the following form: 

dw r c 

(1.6.3) — = J I [Q0 t (0 n - v n ) - pv n ]dS. 

s 
We recall that v n means the normal velocity component of S, and 
n refers to the velocity component of the fluid taken in the direction 
of the normal to S which points outward from R. 

It happens frequently that the boundary surface S of R is made 
up of a number of different pieces which have different properties 
or for which various different conditions are prescribed. Suppose 
first that a portion S P of S is a "physical" boundary containing 
always the same fluid particles. Then n and v n are identical (cf. 
(1.4.2)) and 

S P 
If, in addition, the surface S P is fixed in space, i.e. v n = 0, the 
contribution of S P to dEjdt evidently vanishes, as it should, since no 
energy flows through a fixed boundary containing always the same 
fluid particles. Similarly, the contribution to the energy flux also 
vanishes in the important special case in which S F is a free surface 
on which the pressure p vanishes; this result also accords with what 
one expects on physical grounds. 

Suppose now that S G is a "geometric" surface fixed in space, but 
not necessarily consisting of the same particles of water. In this 
case we have v n = and the flow of energy through S G is given by 



(1.6.4.) 



(1 - 6 - 5) dt 



BASIC HYDRODYNAMICS 15 

dE 



s G 



■ 



O t O n dS. 

s G 

An important special case for us is that in which is the velocity 
potential for a plane progressing wave given, for example, by 

(1.6.6) 0(x,y,z;t) = (p(x — ct,y,z), 

which represents a wave moving with constant velocity c in the 
direction of the cr-axis. The flux through a fixed plane surface S 
orthogonal to the #-axis is easily seen from (1.6.5) to be given by 

(1.6.7) — = - [[ Q c0ldydz. 

s 
The negative sign results since our stipulations amount to saying 
that the region R occupied by the fluid lies on the negative side 
of S (i.e. on the side away from the positive normal, the a?-axis); 
and consequently the energy flux through S due to a progressing 
wave moving in the positive direction of the normal (so that c is 
positive) is such as to decrease the energy in R, as one would expect. 
It is to be noted that there is always a flow of energy through a 
surface S orthogonal to the direction of a progressing wave if 
n =£ — even though the motion of the individual particles of the 
fluid should happen, for example, to be such that the particles move 
in a direction opposite to that of the progressing wave. 

1.7. Formulation of a surface wave problem 

It is perhaps useful — although somewhat discouraging, it must be 
admitted — to sum up the above discussion concerning the fun- 
damental mathematical basis for our later developments by formula- 
ting a rather general, but typical, problem in the hydrodynamics of 
surface waves. The physical situation is indicated in Figure 1.7.1; 
what is intended is a situation like that on any ocean beach. The 
water is assumed to be initially at rest and to fill the space R defined by 

— h(cc, z) 5^ y ^ 0, — oo < z < oo, 

and extending to -f- go in the ^-direction. At the time t = 0, a given 
disturbance is created on the surface of the water over a region D 
(by the wind, perhaps), and one wishes to determine mathematically 
the subsequent motion of the water; in particular, the form of the 



16 



WATER WAVES 




"~-- -1\ , »\ 



Fig. 1.7.1. A very general surface wave problem 



free surface y = r](x, z; t) is to be determined. On the basis of these 
assumptions the following conditions should be satisfied: First of 
all, the differential equation to be satisfied by is, of course, the 
Laplace equation 

( x s (z; t) ^ x < oo 

(1.7.1) V 2 = XX + yy + ZZ = O for - h(x, z)^y ^ rj(x, z; t) 

I — 00 < Z < 00 

It is to be noted that x $ (z; t), the abscissae of the water line on shore, 
and rj(x, z; t), the free surface elevation, are not known in advance 
but are rather to be determined as an integral part of the solution. 
As boundary condition to be satisfied at the bottom of the sea we 
have 

30 

(1.7.2) — - = for y = - h{x, z), 



while the free surface conditions are the kinematic condition (cf. 

(1.4.5)) 

(1.7.3) xVx -0 y + z r] z + Vt = for y = V (x, z; t), 
and the dynamic condition 

(1.7.4) grj+0 t + 1(01 + 01 + 2 z ) = F(x, z; t) on y = r)(x, z; t), 

with F(x, z; t) = everywhere except over the region D where the 
disturbance is created. At oo, i.e. for x -> oo and | z \ -> oo, we 
might prescribe that and r\ remain bounded, or perhaps even that 
they and certain of their derivatives tend to zero. Next we have the 
initial conditions 



(1.7.5; 
(i.7.e; 



r](x, z; t) = for t = 0, 
X = = = o for t = 0, 



BASIC HYDRODYNAMICS 17 

appropriate to the condition of rest in an equilibrium position. 
Finally we must prescribe conditions fixing the disturbance; this 
could be done, for example, by giving the pressure p over the 
disturbed region D of the surface, in other words by prescribing the 
function F in (1.7.4) appropriately there. 

One has only to write down the above formulation of our problem 
to realize how difficult it is to solve it. In the first place the problem 
is nonlinear, but what makes for perhaps even greater difficulties is 
the fact that the free surface is not known a priori and hence the 
domain in which the velocity potential is to be determined is not 
known in advance — aside from the fact that its boundary varies 
with the time. 

These are, however, not the only difficulties in the above problem. 
If we assume that the function is regular throughout the interior 
of R and uniformly bounded (together with some of its derivatives, 
perhaps) in R, the formulation of the problem given above would 
seem to be reasonable from the point of view of mechanics. However, 
the solution would probably not exist for all Z > for the following 
reason: everyone who has visited an ocean beach is well aware that 
the waves do not come in smoothly all the way to the shore (except 
possibly in very calm weather), but, rather, they steepen in front, 
curl over, and eventually break. In other words, any mathematical 
formulation of the problem which would fit the commonly observed 
facts even for a limited time would necessitate postulating the 
existence of singularities of unknown location in both space and time. 

Because of the difficulty of the general nonlinear theory very little 
progress has been made in solving concrete problems which employ 
it. An exception is the problem of proving the existence of two- 
dimensional periodic progressing waves in water of uniform depth. 
This was done first by Nekrassov [N.l], [N.la] and by Levi-Civita 
[L.7] for water of infinite depth, and later by Struik [S.29] for water of 
constant finite depth. In Chapter 12 an account of Levi-Civita's theory 
is given. In both cases the authors prove rigorously the existence of 
waves having amplitudes near to zero by showing that perturbation 
series in the amplitude converge. Another exception to the above 
statement is the problem of the solitary wave, the existence of which, 
from the mathematical point of view, has been proved recently by 
Lavrentieff [L.4] and by Friedrichs and Hyers [F.13]; an account of 
the work of the latter two authors is given in Chapter 10.9. 

It seems likely that solutions of problems in the full nonlinear 



18 WATER WAVES 

version of the theory will, for a long time to come, continue to be 
of the nature of existence theorems for motions of a rather special 
nature. 

In order to make progress with the theory of surface waves it is 
in general necessary to simplify the theory by making special hypoth- 
eses of one kind or another which suggest themselves on the basis 
of the general physical circumstances contemplated in a given class 
of problems. As we have already explained in the introduction, up 
to now attention has been concentrated almost exclusively upon the 
two approximate theories which result when either a) the amplitude 
of the surface waves is considered small (with respect to wave length, 
for example), or b) the depth of the water is considered small (again 
with respect, say, to wave length). The first hypothesis leads to a 
linear theory and to boundary value problems more or less of classical 
type; while the second leads to a nonlinear theory for initial value 
problems, which in lowest order is of the type employed in wave 
propagation in compressible gases. If both hypotheses are made, the 
result is a linear theory involving essentially the classical linear 
wave equation; the present theory of the tides belongs in this class 
of problems. 

In the next chapter we derive the approximate theories arising 
from the two hypotheses by starting from the general theory and 
then developing formally with respect to an appropriate parameter — 
essentially the surface wave amplitude in one case and the depth 
of the water in the other — and in subsequent chapters we continue 
by treating a variety of special problems in each of the two classes. 



CHAPTER 2 



The Two Basic Approximate Theories 

2.1. Theory of waves of small amplitude 

It has already been stated that the theory of waves of small 
amplitude can be derived as an approximation to the general theory 
presented in Chapter I on the basis of the assumption that the 
velocity of the water particles, the free surface elevation y=r)(x, z; t), 
and their derivatives, are all small quantities. We assume, in fact, 
that the velocity potential and the surface elevation rj possess the 
following power series expansions with respect to a parameter e: 

(2.1.1) = e0 (l) + e 2 {2) + e z (z) + . . ., and 

(2.1.2) rj(x, z; t)=rj w (x, z; t)+erj (1) (x, z; t)+e 2 r] {2) (x, z; t)+ . . .. 

It follows first of all that each of the functions {k) (x, y, z; t) is 
a solution of the Laplace equation, i.e. 

(2.1.3) \/20(k) = 

We turn next to the discussion of the boundary conditions. At a 
fixed physical boundary (cf. section 1.4) of the fluid we have clearly 
the conditions 

d0W 

(2.1.4) — = 0, 

on 

in which djdn represents differentiation along the normal to the 
boundary surface. 

At a free surface S: y = r)(x, z; t) on which the pressure is zero we 
have two boundary conditions. One of them arises from the Bernoulli 
law and has the form 

gr] +0 t + \{®l + #; + 0\) = on S. 
Upon insertion of (2.1.1) and (2.1.2) in this condition and developing 
t , 0% etc. systematically in powers of s (due regard being paid to 
the fact that the functions ( t k) , & { ®, etc. are to be evaluated for 

19 



20 WATER WAVES 

y = n(x, z; t) and that r\ in its turn is given in terms of e by 
yj = ri (Q) + erj (1) + . . .) one finds readily the conditions 

(2.1.5) rj (0) = 0, 

(2.1.6) #7<D + 0? = 0, 

(2.1.7) g?7< 2 > +0f + J[((pW)i + (0W)» + (<Z>W)2] +V*J'^W = 



to be satisfied for ?/ = r) i0) , and since ?y (0) == from (2.1.5) it follows 
that the conditions (2.1.6), (2.1.7), etc. are all to be satisfied on the 
originally undisturbed surface of the water y = 0. The other boundary 
condition on S arises from the fact that the water particles stay 
on S (cf. section 1.4); it is expressed in the form 

&.V» +® z Vz +rjt = ®y on S. 

Insertion of the power series for and rj in this expression leads to 
the conditions 

(2.1.8) ,{« = o, 

(2.1.9) *»l,»+«? , l»»+lfc W =.*«, 

(2.1.10) 0<M°> + &S? + n f = <t>f - *»i,» - aW 

V \*^xy Vx \ ^zy Vz ^yy /» 



which are also to be satisfied for y = 0. 

In view of the fact that r] {0) = 0, the free surface conditions can 
be put in the form 

(2.1.11) #7<» + 0® =0, 

(2.1.12) grjM + 0« = - K^?) 1 + (O 2 + (^) 2 ] - V tn *£> 



(2.1.13) g?y(»> +0j n) = JWn-l), 

in which the symbol 2r(n-i) refers to a certain combination of the 
functions ^ (fc) and (k) with fc ^ n — 1, and all conditions are to be 
satisfied for y = 0. Similarly, the other set of free surface conditions 
becomes 



THE TWO BASIC APPROXIMATE THEORIES 21 

(2.1.14) *!?=*?, 

(2.1.15) ^ = Of - *«,£> - *<V> + *£,y«, 

(2.1.16) ^ n) = 0i w) + G^-D, 

in which G (n_1) depends only upon functions ^ (&) and (fc) with 
& <£ n-— 1, and once more all conditions are to be satisfied for 
y = 0. This theory therefore is a development in the neighborhood 
of the rest position of equilibrium of the water. 

The relations (2.1.11) to (2.1.16) thus, in principle, furnish a means 
of calculating successively the coefficients of the series (2.1.1) and 
(2.1.2), assuming that such series exist: The conditions (2.1.11) and 
(2.1.14) at the free surface together with appropriate conditions at 
other boundaries, and initial conditions for t = 0, would in conjunc- 
tion with V 2 (1) = lead to unique solutions tj (1) and (1) . Once 
r] (1) and {1) are determined, they can be inserted in the conditions 
(2.1.12) and (2.1.15) to yield two conditions for r) {2) and & {2) which 
with the subsidiary boundary and other conditions on & {2) serve to 
determine them, etc. One could interpret the work of Levi-Civita [L.7] 
and Struik [S.29] referred to in section 1.7 as a method of proving 
the existence of progressing waves which are periodic in x by showing 
that the functions and r\ can indeed be represented as convergent 
power series in s for s sufficiently small. 

In what follows in Part II of this book we shall content ourselves 
in the main with the degree of approximation implied in breaking 
off the perturbation series after the terms s0 {1) and etj {1) in the series 
(2.1.1) and (2.1.2), i.e. we set = e0 {1) and r\ =er) {1) . With this 
stipulation the free surface conditions (2.1.11) and 2.1.14) yield 

(2.1.17) gr] +0 t = o] 

I for y = 0. 

(2.1.18) m -0 y = o\ 

By elimination of r\ between these two relations the single condition 
on 0: 

(2.1.19) u J rg y = o fory = 

is obtained; this condition is the one which will be used mainly in 
Part II in order to determine from V 2 = 0, after which the free 
surface elevation r\ can be determined from (2.1.17). The usual 
method of obtaining the last three conditions is to reject all but 



22 WATER WAVES 

the linear terms in y] and and their derivatives in the kinematic 
(cf. (1.4.5)) and dynamic (cf. (1.4.6)) free surface boundary con- 
ditions. By proceeding in this way we can obtain a first approximation 
to the pressure p (which was not considered in the above general 
perturbation scheme) in the form: 

(2.1.20) -= -gy-0 t . * 

Q 

We can now see the great simplifications which result through 
the linearization of the free surface conditions: not only does the 
problem become linear, but also the domain in which its solution 
is to be determined becomes fixed a priori and consequently the 
surface wave problems in this formulation belong, from the mathe- 
matical point of view, to the classical boundary problems of potential 
theory. 

2.2. Shallow water theory to lowest order. Tidal theory 

A different kind of approximation from the foregoing linear theory 
of waves of small amplitude results when it is assumed that the 
depth of the water is sufficiently small compared with some other 
significant length, such as, for example, the radius of curvature of 
the water surface. In this theory it is not necessary to assume that 
the displacement and slope of the water surface are small, and the 
resulting theory is as a consequence not a linear theory. There are 
many circumstances in nature under which such a theory leads to 
a good approximation to the actual occurrences, as has already been 
mentioned in the introduction. Among such occurrences are the tides 
in the oceans, the "solitary wave" in sufficiently shallow water, and 
the breaking of waves on shallow beaches. In addition, many pheno- 
mena met with in hydraulics concerning flows in open channels such 
as roll waves, flood waves in rivers, surges in channels due to sudden 
influx of water, and other kindred phenomena, belong in the nonlinear 
shallow water theory. Chapters 10 and 11 are devoted to the working 
out of consequences of the shallow water theory. 

The shallow water theory is, in its lowest approximation, the basic 
theory used in hydraulics by engineers in dealing with flows in open 

* In case the surface pressure p (x, z; t) is not zero one finds readily that (2.1.17) 
is replaced by 

(2.1.20)! grj + t== -pjQ, 

while (2.1.18) remains unaltered. 



THE TWO BASIC APPROXIMATE THEORIES 23 

channels, and also the theory commonly referred to in the standard 
treatises on hydrodynamics as the theory of long waves. We begin by 
giving first a derivation of the theory for two-dimensional motion 
along essentially the lines followed by Lamb [L.3], p. 254. As usual, 
the undisturbed free surface of the water is taken as the #-axis and 
the y-axis is taken vertically upwards. The bottom is given by 
y = — h(x), so that h represents the variable depth of the undisturbed 
water. The surface displacement is given by y = rj(x, t). The velocity 
components are denoted by u(x, y, t) and v(x, y, t). 
The equation of continuity is 

(2.2.1) u x +v y = 0. 

The conditions to be satisfied at the free surface are the kinematical 
condition: 

(2.2.2) ( m + urj x - v) \ y=rj = 0; 
and the dynamical condition on the pressure: 

(2.2.3) PV„ = 0. 
At the bottom the condition is 

(2.2.4) (uh x +v)\ y= _ h = 0. 
Integration of (2.2.1) with respect to y yields 

(2.2.5) f (u x )dy+v\\ = Q. 

J —h 

Use of the condition (2.2.2) at y = r\ and (2.2.4) at y = — h yields 
the relation 



(2.2.6) f u x dy + rj t +u\ n "n x +u\_ h -h a 

J —h 



0. 



We introduce the relation 

a 



( 2 - 2 - 7 ) al udy = u \ y=n • rj x + u \ y= _ h ■ h x + u x dy. 

x J -Hx) J -h 



-h(x) 

and combine it with (2.2.6) to obtain 
(2.2.8) lj\dy=-r lt . 

Up to this point no approximations have been introduced. 

The shallow water theory is an approximate theory which results 
from the assumption that the ^/-component of the acceleration of 



24 WATER WAVES 

the water particles has a negligible effect on the pressure p, or, 
what amounts to the same thing, that the pressure p is given as in 
hydrostatics * by 

(2.2.9) p = g Q (rj - y). 

The quantity q is the density of the water. A number of consequences 
of (2.2.9) are useful for our purposes. To begin with, we observe that 

(2.2.10) p x = gerj x , 

so that p x is independent of y. It follows that the ^-component of 
the acceleration of the water particles is also independent of y; 
and hence u, the ^-component of the velocity, is also independent 
of y for all t if it was at any time, say at t = 0. We shall assume 
this to be true in all cases — it is true for example in the important 
special case in which the water was at rest at t = — so that u=u(x, t) 
depends only on x and t from now on. As equation of motion in the 
^-direction we may write, therefore, in view of (2.2.10): 

(2.2.11) u t +uu x = - gr\ x . 

This is simply the usual equation of motion in the Eulerian form, 
use having been made of u y = 0. In addition, (2.2.8) may now be 
written 

(2.2.12) [u(r] + h)] x = - Vt , 

rv rv 

since udy = u\ dy on account of the fact that u is independent 

J — h J — h 

of y. The two first order differential equations (2.2.11) and (2.2.12) 
for the functions u(x, t) and rj(x, t) are the differential equations of 
the nonlinear shallow water theory. Once the initial state of the fluid 
is prescribed, i.e. once the values of u and rj at the time t = are 
given, the equations (2.2.11) and (2.2.12) yield the subsequent 
motion. 

If in addition to the basic assumption of the shallow water theory 
expressed by (2.2.9) we assume that u and rj, the particle velocity 
and free surface elevation, and their derivatives are small quantities 
whose squares and products can be neglected in comparison with 
linear terms, it follows at once that equations (2.2.11) and (2.2.12) 
simplify to 

(2.2.13) u t = -grj xi 

(2.2.14) (uh) x = - rj t , 

* We have p y = —go and (2.2.9) results through the use of p = for y = rj. 



THE TWO BASIC APPROXIMATE THEORIES 25 

from which r\ can be eliminated to yield for u the equation 

1 

(2.2.15) (uh) xx --u tt = 0. 

If, in addition, the depth h is constant it follows readily that u 
satisfies the linear wave equation 

1 

(2.2.16) u xx -—u u = 0. 

In this case r\ satisfies the same equation. One observes therefore 
the important result that the propagation speed of a disturbance is 
given by Vgh. In principle, this linearized version of the shallow water 
theory is the one which has always been used as the basis for the 
theory of the tides. Of course, the tidal theory for the oceans requires 
for its complete formulation the introduction of the external forces 
acting on the water due to the gravitational attraction of the moon 
and the sun, and also the Coriolis forces due to the rotation of the 
earth, but nevertheless the basic fact about the tidal theory from 
the standpoint of mathematics is that it belongs to the linear shallow 
water theory. The actual oceans do not from most points of view 
impress one as being shallow; in the present connection, however, 
the depth is actually very small compared with the curvature of 
the tidal wave surface so that the shallow water approximation is an 
excellent one. That the tidal phenomena should be linear to a good 
approximation would also seem rather obvious on account of the 
small amplitudes of the tides compared with the dimensions of the 
oceans. A few additional remarks about tidal theory and some other 
applications of the linearized version of the shallow water theory to 
concrete problems (seiches in lakes, and floating breakwaters, for 
example) are given in Chapter 10.13. 

2.3. Gas dynamics analogy 

It is possible to introduce a different set of dependent variables 
in such a way that the equations of the shallow water theory become 
analogous to the fundamental differential equations of gas dynamics for 
the case of a compressible flow involving only one space variable x. 
(This seems to have been noticed first by Riabouchinsky [R.8].) 
To this end we introduce the mass per unit area given by 

(2.3.1) Q = Q(rj+h). 



26 WATER WAVES 

Since h depends only on x we have 

(2.3.2) Qt = QVf 

We next define the force p per unit width: 

(2.3.3) p= C pdy, 

J —h 

which, in view of (2.2.9) and (2.3.1), leads to 

(2.3.4) V = ^(ri+hY=~~Q 2 - 

The relation between p and g is thus of the form p = Ag y with 
y = 2, that is, the "pressure" p and the "density" g are connected 
by an "adiabatic" relation with the fixed exponent 2. 
Equation (2.2.11) may now be written 

q(t} + h)(u t + uu x ) = - gg(r] + h)r\ x 

and this, in turn, may be expressed through use of (2.3.1) and 

(2.3.4) as follows: 

(2.3.5) g(u t + uu x ) = — p x + gqh x , 

as one can readily verify. 

The equation (2.2.12) may be written as 

(2.3.6) (Qu) x = -Q t , 

in view of (2.3.2) as well as (2.3.1). The differential equations (2.3.5) 
and (2.3.6), together with the "adiabatic" law p = gQ 2 /2g given by 
(2.3.4), are identical in form with the equations of compressible gas 
dynamics for a one-dimensional flow except for the term ggh x on 
the right hand side of (2.3.5), and this term vanishes if the original 
undisturbed depth h of the water is constant. The "sound speed" c 
corresponding to our equations (2.3.5) and (2.3.6) is, in analogy with 
gas dynamics, given by c = Vdp/d^, and this from (2.3.4) and (2.3.1) 
has the value 

(2.3.7) c = )/^ = Vgfo+A). 

It will be seen later that c{x, t) represents the local speed at which 
a small disturbance advances relative to the water. 



THE TWO BASIC APPROXIMATE THEORIES 27 

2.4. Systematic derivation of the shallow water theory 

It is of course a matter of importance to know under what cir- 
cumstances the shallow water theory can be expected to furnish 
sufficiently accurate results. The only assumption made above in 
addition to the customary assumptions of hydrodynamics was that 
the pressure is given as in hydrostatics by (2.2.9), but no assumption 
was made regarding the magnitude of the surface elevation or the 
velocity components. Consequently the shallow water theory may 
be accurate for waves whose amplitude is not necessarily small, 
provided that the hydrostatic pressure relation is not invalidated. 
The above derivation of the shallow water theory is, however, open 
to the objection that the role played by the undisturbed depth of the 
water in determining the accuracy of the approximation is not put 
in evidence. In fact, since we shall see later on that all motions die 
out rather rapidly in the depth, it would at first sight seem reasonable 
to expect that the hydrostatic law for the pressure would be, on the 
whole, more accurate the deeper the water. That this is not the 
case in general is well known, since the solutions for steady progressing 
waves of small amplitude (i.e. for solutions obtained by the linearized 
theory) in water of uniform but finite depth are approximated 
accurately by the solutions of the shallow water theory (when it also 
is linearized) only when the depth of the water is small compared 
with the wave length (cf. Lamb [L.3], p. 368). It is possible to give 
a quite different derivation of the shallow water theory in which the 
equations (2.2.11) and (2.2.12) result from the exact hydrodynamical 
equations as the approximation of lowest order in a perturbation 
procedure involving a formal development of all quantities in powers 
of the ratio of the original depth of the water to some other 
characteristic length associated with the horizontal direction.* The 
relation (2.2.9) is then found to be correct within quadratic terms 
in this ratio. In this section we give such a systematic derivation 
of the shallow water theory, following K. O. Friedrichs (see the 
appendix to [S.19]), which, unlike the derivation given in section 

* In this book the parameter of the shallow water theory is defined in two 
different ways: in dealing with the breaking of waves in Chapter 10.10 it is 
the ratio of the depth to a significant radius of curvature of the free surface; 
in dealing with the solitary wave, however, it is essentially the ratio of the depth 
to the quantity U 2 jg, with U the propagation speed of the wave, and in this 
case the development is carried out for U 2 /gh near to one. In still other 
problems it might well be defined differently in terms of parameters that are 
characteristic for such problems. 



28 



WATER WAVES 



2.2 above, is capable of yielding higher order approximations. 
The disposition of the coordinate axes is taken in the usual manner, 
with the x, z-plane the undisturbed water surface and the y-axis 
positive upward. The free surface elevation is given by y = yj{x, z, t) 
and the bottom surface by y = — h(x, z). We recapitulate for the 
sake of convenience the differential equations and boundary con- 
ditions in terms of the Euler variables, that is, the equations of 
continuity and motion, the vanishing of the rotation, and the boundary 
conditions: 



(2.4.1) 


u x +v y +w s = 0, 




1 

u t + uu x + vu y + wu z = — — p x 

Q 


(2.4.2) 


1 
v t + uv x + VVy + wv z = — — p y 




1 

W t + Wffij, + VWy + ww z = — — p z 
I Q 


(2.4.3) 


W y = V z , U g = W xi V x = Uy, 


(2.4.4) 


Vt + u Vx + wrjz = v at y = rj, 


(2.4.5) 


p = o at y = rj, 


(2.4.6) 


uh x + v + wh z = at y = — 



We now introduce dimensionless variables through the use of two 
lengths d and k, with d intended to represent a typical depth and k 
a typical length in the horizontal direction — it is characteristic of 
the procedure followed here that the horizontal and vertical direc- 
tions are not treated in the same way. The new independent variables 
are as follows: 

(2.4.7) x = cc/k, y = y/d, z = z/k, r = t Vgd/k, 

while the new dimensionless dependent variables are 



(2.4.8; 



p 






(kVgd/d)- 1 v, w = (Vgd)- 1 w 



{rj — rjfd, h = h/d. 
In addition, we introduce the important parameter 
(2.4.9) g = d 2 /k 2 



THE TWO BASIC APPROXIMATE THEORIES 29 

in terms of which all quantities will be developed; when this parameter 
is small the water is considered to be shallow. This means, of course, 
that d is small compared with k, and hence that the x and z coor- 
dinates (cf. (2.4.7)) are stretched differently from the y coordinate 
and in a fashion which depends upon the development parameter. 
Since it is the horizontal coordinate which is strongly stretched 
relative to the depth coordinate, it seems reasonable to refer to the 
resulting theory as a shallow water theory. The stretching process 
combined with a development with respect to a is the characteristic 
feature of what we call the shallow water theory throughout this 
book. The dimensionless development parameter a has a physical 
significance, of course, but its interpretation will vary depending 
on the circumstances in individual cases, as has already been noted 
above. For example, consider a problem in which the motion is to be 
predicted starting from rest with initial elevation y = n (x, y, z) 
prescribed; from (2.4.8) we have 

y = vo = dfj (x, y> *) 



l x y z\ 



from which we obtain 



d 2 



It is natural to assume that the dimensionless second derivative 
Voxx will be at least bounded and consequently one sees that the 
assumption that a is small might be interpreted in this case as 
meaning that the product of the curvature of the free surface of the 
water and a typical depth is a small quantity. 

The object now is to consider a sequence of problems depending 
on the small parameter a and then develop in powers of a. Introduc- 
tion of the new variables in the equations (2.4.1) to (2.4.6) yields 

(2.4.1)' ou x +v v + ow z = 0, 

G[U t + uu x + wu z + p x ] + vu y = 0, 
(2.4.2)' | a [v t + uv x + wv z -f p y + 1] + vv y = 0, 

. o[w t + uw x + ww z + p z ] + vw y = 0, 
(2.4.3)' w y = v zi u z = w x , v x = u y , 

(2.4.4)' o[rj t + utj x + wrj z ] = v at y = n. 



30 



WATER WAVES 



(2.4.5)' 
(2.4.6)' 



p = o at y ■ = TJ, 

o[uh x + wh g ] + v = at y 



when bars over all quantities are dropped and x is replaced by t. 
The next step is to assume power series developments for u, v, 
w, r], and p: 



(2.4.10) 



u = u 



(0) 



V = V 



(0) 



+ u a) a + u 
,(i) 



+ v^g + U< z ><7 a + 



(2)^2 



+ 



zc 



(0) 



+ o^c + w™o* + 



(2)^-2 



^<U) _|_ jy 



(1), 



^(2)^2 _|_ 



,(0) 



+ p U) (7 + p 



(2)^2 



+ 



and insert them in the equations (2.4.1)' to (2.4.6)' to obtain, by 
equating coefficients of like powers of g, equations for the successive 
coefficients in the series, which are of course functions of x, y, z, 
and t. The terms of zero order yield the equations 



(2.4.1 ) 
(2.4.2); 

(2.4.3); 
(2.4.4); 
(2.4.5); 
(2.4.6); 



,(0) 



(0) 



V (0) W W = 



v w u y 



<r 



(0) 



D (0) U {0) 

7,(0) 



,(0) 



,(0) 



at y 
at y 
at y 



V {0) - 

™(0) 
'/ ' 

?7 (0) , 
- h. 



,(0) 



These equations yield the following: 



(2.4.11) 
(2.4.12) 
(2.4.13) 
(2.4.14) 



zv 



,(0) 



(0) _ M ,(0) 



= w {0) (x, z, t 



M (0) = u < 0) {x s z,t), 
p {0) (x,rj {0) ,z, t) = 0, 



which contain the important results that the vertical velocity com- 
ponent is zero and the horizontal velocity components are independent 
of the vertical coordinate y in lowest order. 

The first order terms arising from (2.4.1)' to (2.4.6)' in their turn 



THE TWO BASIC APPROXIMATE THEORIES 31 

yield the equations 

(2.4.1); y«+w» = - -*« 

/ ^i 0) + u^uf + ^<°)^ 0) + pi 0) = 0, 
(2.4.2); V { y 0) +1=0, 

' w\ 0) + M<°>Kji 0) + W^W { Z 0) + pi 0) - 0, 

(2.4.4); rjf ] + w (0) ^ 0) + wWrjW = z> (1) at t/ = ??< 0) , 

(2.4.6); w (0) /* x + w (0) /* 2 + u (1) =0aty=-A, 

upon making use of (2.4.11), (2.4.12), and (2.4.13). Equation (2.4.1); 
can be integrated at once since u (0) and w {0) are independent of y 
to yield 

(2.4.15) d<« = - (i4 0) + wf ] )y + F{x, *, t), 

with F an arbitrary function which can be determined by using 
(2.4.6);; the result for v {1) is then 

(2.4.16) *<« = - (uf + ^ 0) )t/ - [(w< 0) /*)* + (wm) z ] y= _ n . 

To second order the vertical component of the velocity is thus linear 
in the depth coordinate. In similar fashion the second of the equations 
(2.4.2); can be integrated and the additive arbitrary function of 
x, z, t determined from (2.4.14); the result is 

(2.4.17) p<°>0z, y, z, t) = rjM(x, z, t) - y 

which is obviously the hydrostatic pressure relation (in dimension- 
less form). 

In the derivation of the shallow water theory given in the preceding 
section this relation was taken as the starting point; here, it is 
derived as the lowest order approximation in a formal perturbation 
scheme. However, it is of course not true that we have proved that 
(2.4.17) is in some sense an appropriate assumption: instead, it 
should be admitted frankly that our dimensionless variables were 
introduced in just such a way that (2.4.17) would result. If it could 
be shown that our perturbation procedure really does yield a correct 
asymptotic development (that the development converges seems 
unlikely since the equations (2.4.1)' to (2.4.6)' degenerate in order 
so greatly for g = 0) then the hydrostatic pressure assumption could 
be considered as having been justified mathematically. A proof that 
this is the case would be of great interest, since it would give a 
mathematical justification for the shallow water theory; to do so in 



32 WATER WAVES 

a general way would seem to be a very difficult task, but Friedrichs 
and Hyers [F.13] have shown that the development does yield 
the existence of the solution in the important special case of the 
solitary wave (cf. Chapter 10.9). (Keller [K.6] had shown earlier 
that the formal procedure yields the solitary wave.) The problem 
is of considerable mathematical interest also because of the following 
intriguing circumstance: the approximation of lowest order to the 
solution of a problem in potential theory is sought in the form of a 
solution of a nonlinear wave equation, and this means that the 
solution of a problem of elliptic type is approximated (at least in the 
lowest order) by the solution of a problem of hyperbolic type. 

The values of v (l) and p (0) given by (2.4.16) and (2.4.17) are now 
inserted in the first and third equations of (2.4.2 )^ and in (2.4.4)/ 
to yield finally 

(2.4.18) w| 0) + tt«°)u? + w^u^ + w<°> = 0, 

(2.4.19) w[ 0) + u {0) w ( x 0) + w {0) w ( z 0) + ^ 0) = 0, 

(2.4.20) rjf + [w<°>(y°> + h)] x + [w<®>(V°> + h)] z = 0, 

as definitive equations for w (0) , w (0) , and rj i0) — all of which, we repeat, 
depend only upon cc, z, and t. If the superscript is dropped, w {0) is 
taken to be zero, and it is assumed that all quantities are independent 
of z, one finds readily that these equations become identical with 
equations (2.2.11) and (2.2.12) except for the factor g in (2.2.11) 
which is missing here because of our introduction of a dimensionless 
pressure. 

It is clear that the above process can be continued to obtain the 
higher order approximations. An example of such a calculation will 
be given later in Chapter 10.9, where we shall see that the first non- 
trivial term in the development which yields the solitary wave is of 
second order. 



PART II 






Summary 



In Part II we treat a variety of problems in terms of the theory 
which arises through linearization of the free surface condition (cf. 
the preceding chapter); thus the problems refer to waves of small 
amplitude. To this theory the names of Cauchy and Poisson are 
usually attached. The material falls into three different types, or 
classes, of problems, as follows: A) Waves that are simple harmonic 
in the time. These problems are treated in Chapters 3, 4, and 5 and 
they include a study of the classical standing and progressing wave 
solutions in water of uniform depth, and waves over sloping beaches 
and past obstacles of one kind or another. The mathematical tools 
employed here comprise, aside from classical methods in potential 
theory, a thorough-going use of integrals in the complex domain. 
B) Waves created by disturbances initiated at an instant when the water 
is at rest. These problems, which are treated in Chapter 6, comprise 
a variety of unsteady motions, including the propagation of waves 
from a point impulse and from an oscillatory source. Uniqueness 
theorems for the unsteady motions are derived. The principle mathe- 
matical tool used in solving these problems is the Fourier transform. 
The method of stationary phase is justified and used. C) Waves 
arising from obstacles immersed in a running stream. This category 
of problems differs from the first two in that the motion to be in- 
vestigated is a small oscillation in the neighborhood of a uniform 
flow, while the former cases concern small oscillations near the state 
of rest. This difference is in one respect rather significant since the 
problems of the first two types require no restriction on the shape of 
immersed bodies, or obstacles, while the third type of problem 
requires that the immersed bodies should be in the form of thin disks, 
since otherwise the flow velocity would be changed by a finite amount, 
and a linearization of the free surface condition would not then be 
justified. In other words, the problems of this third type require 

35 



36 WATER WAVES 

a linearization based on assuming a small thickness for any immersed 
bodies, as well as a linearization with respect to the amplitude of the 
surface waves. These problems are treated in Chapters 7, 8, and 9. 
The classical case of the waves created by a small obstacle in a running 
stream of uniform depth is first treated. This includes the classical 
shipwave problem, discussed in Chapter 8, in which the "ship" is 
treated as though it could be replaced by a point singularity. A 
treatment is given in Chapter 9 of the problem of the waves created 
by a ship moving through a sea of arbitrary waves, assuming the 
ship to be a floating rigid body with six degrees of freedom and with 
its motion determined by the propeller thrust and the pressure of the 
water on its hull. 

Finally, in an Appendix to Part II a brief summary of some of 
the more recent literature concerned with the above types of problems 
is given, since the cases selected for detailed treatment here do not 
by any means exhaust the interesting problems which have been 
solved. 



SUBDIVISION A 

WAVES SIMPLE HARMONIC IN THE TIME 

CHAPTER 3 

Simple Harmonic Oscillations in Water 
of Constant Depth 



3.1. Standing waves 

In Chapter 2 we have derived the basic theory of irrotational 
waves of small amplitude with the following results (in the lowest 
order, that is). Assuming the x, 2-plane to coincide with the free 
surface in its undisturbed position, with the y-axis positive upward, 
the velocity potential 0(x, y, z; t) satisfies the following conditions: 

(3.1.1) V 2 <Z> = XX + yy +0 ZZ = O 

in the region bounded above by the plane y = and elsewhere by 
any other given boundary surfaces. The free surface condition under 
the assumption of zero pressure there is 

(3.1.2) tt +g0 v = O for y = 0. 

The condition at fixed boundary surfaces is that d&jdn = 0; for 
water of uniform depth h = const, we have therefore the condition 

(3.1.3) y = for y = — h. 

Once the velocity potential has been determined the elevation 
Y](x, z; t) of the free surface is given by 

(3.1.4) n = - l t (x, 0, z; t). 



Conditions at go as well as appropriate initial conditions at t = 
must also be prescribed. 

In this section we are interested in those special types of standing 
waves which are simple harmonic in the time; we therefore write 

37 



38 WATER WAVES 

(3.1.5) 0(x, y, z; t) = e iat <p(x, y, z) * 

with cp a real function, and with the understanding that either the 
real or the imaginary part of the right hand side is to be taken. 
The problems to be treated here thus belong to the theory of small 
oscillations of dynamical systems in the neighborhood of an equilib- 
rium position. 

The conditions on given above translate into the following 
conditions on cp: 

(3.1.6) V 2 cp = 0, — h < y < 0, — oo < x, z < oo, 

o 2 

(3.1.7) cp y - ~(jp = 0, y = 0, 

(3.1.8) <p v = 0, y = - h. 

As conditions at oo we assume that cp and cp y are uniformly bounded.** 
Arbitrary initial conditions cannot now be prescribed, of course, 
since we have assumed the behavior of our system to be simple 
harmonic in the time. The free surface elevation is given by 

icj 

(3.1.9) r) = e iat • cp{x, 0, z). 

We look first for standing wave motions which are two-dimensional, 
so that (p depends only upon x and y: 99 = (p(x, y), and also consider 
first the case of water of infinite depth, i.e. h = 00. One verifies 
readily that the functions 

(3.1.10) I V = emVcosmx 

\ <p = e my sin mx 

are harmonic functions which satisfy the free surface condition 
(3.1.7) provided that the constant m satisfies the relation 

(3.1.11) m = G 2 /g. 

In addition, the conditions at 00 are satisfied. In particular, it is of 
interest to observe that the oscillations die out exponentially in the 
depth. The free surface elevation is then given by 

* The most general standing wave would be given by = f(t)<p(x, y, z). This 
means, of course, that the shape of the wave in space is fixed within a multiplying 
factor depending only on the time. Thus nodes, maxima and minima, etc. occur 
at the same points independent of the time. 

** This means that the vertical components of the displacement and velocity 
are bounded at 00. One could prescribe more general conditions at co without 
impairing the uniqueness of the solutions of our boundary value problems, but 
it does not seem worth while to do so in this case. 



SIMPLE HARMONIC OSCILLATIONS 39 

ia . , ( cos mx 

(3.1.12) w = — — e 2<rt • 

g { sin ma? 

It should be pointed out specifically that our boundary problem, 
though it is linear and homogeneous, has in addition to the solution 
(p = o a two-parameter set of "non trivial" solutions obtained by 
taking linear combinations of the two solutions given in (3.1.10). 
The surface waves given by (3.1.10) are thus simple harmonic in 
x as well as in t. The relation (3.1.11) furthermore states the very 
important fact that the wave length X given by 

(3.1.13) I = Znjm = 2nglo 2 

is not independent of the frequency of the oscillation, but varies 
inversely as its square. 

The above discussion yields standing wave solutions of physically 
reasonable type, but one nevertheless wonders whether there might 
not be others— for example, standing waves which are not simple 
sine or cosine functions of x, but rather waves with amplitudes which, 
for example, die out as x tends to infinity. Such waves do not occur 
in two dimensions,* however, in the sense that all solutions for water 
of infinite depth, except 92 = 0, of the homogeneous boundary problem 
formulated in (3.1.6) and (3.1.7) together with the condition that 99 
and (p y are uniformly bounded at 00 are given by (3.1.10) with m 
satisfying (3.1.11). This is a point worth pausing to prove, especially 
since the method of proof foreshadows a mode of attack on our 
problems which will be used in a more essential way later on. The 
first step in the uniqueness proof is to introduce the function yi(x, y) 
defined by 

(3.1.14) y) = (p y — mcp, m > 0. 

Since cp is a harmonic function, obviously xp is also a harmonic func- 
tion. In addition, \p vanishes for y = on account of its definition 
and (3.1.7). Hence ip can be continued by reflection over the a>axis 
into a potential function which is regular and defined as a single- 
valued function in the entire x, y-plsme. Since 99 and cp y were assumed 
to be uniformly bounded in the entire lower half plane it follows that 
\p is bounded in the entire x, ?/-plane since reflection in the a?-axis 
does not destroy boundedness properties. Thus yj is a potential func- 
tion which is regular and bounded in the entire x, ?/-plane. By Liou- 

* This statement is not valid in three dimensions as we shall see later on in 
this section. 



40 WATER WAVES 

ville's theorem it is therefore a constant, and since ip = for y = 0, 
the constant must be zero. Hence ip vanishes identically. From (3.1.14) 
it therefore follows that any solutions y(x, y) of our boundary 
value problem are also solutions of the differential equation 

(3.1.15) (p y — mcp = 0, — oo < y < 0. 

The most general solution of this differential equation is given by 

(3.1.16) cp = c(x)e™y 

with c(x) an arbitrary function of x alone. However, cp(x, y) is a 
harmonic function and hence c(x) is a solution of 

d 2 c 

(3.1.17) — + m 2 c = 

dec 2 

which, in turn, has as its general solution the linear combinations 
of sin mx and cos mx. It follows, therefore, that the standing wave 
solutions of our problem are indeed all of the form Ae mv cos (mx+a.),* 
with a and A arbitrary constants fixing the "phase" and the amplitude 
of the wave, while m is a fixed constant which determines the wave 
length A in terms of the given frequency a through (3.1.13). 

In water of uniform finite depth h it is also quite easy to obtain 
two-dimensional standing wave solutions of our boundary value 
problem. One has, corresponding to the solutions (3.1.10) for water of 
infinite depth, the harmonic functions 

\ w = cosh m(y + h) cos mx, 

(3.1.18) \v vy ^ / 

[(p = cosh m(y + h) sin mx, 

as solutions which satisfy the boundary condition at the bottom, 
while the free surface condition is satisfied provided that the con- 
stant m satisfies the relation 

(3.1.19) g 2 = gm tanh mh 

instead of the relation (3.1.11), as one readily sees. Since tanh mh^l 
as h -> oo it is clear that the relation (3.1.19) yields (3.1.11) as limit 
relation for water of infinite depth. The uniqueness of the solutions 
(3.1.18) for the two-dimensional case under the condition of boun- 
dedness at oo was first proved by A. Weinstein [W.7] by a method 

* It can now be seen that the negative sign in the free surface condition (3.1.7) 
is decisive for our results: if this sign were reversed one would find that the 
solution <p analogous to (3.1.16) would be bounded at oo only for c(x) = 0, 
because (3.1.16) would now be replaced by c(x)e- mv , with m > 0. 






SIMPLE HARMONIC OSCILLATIONS 41 

different from the method used above for water of infinite depth 
which can not be employed in this case (see [B. 12]). 

It is of interest to calculate the motion of the individual water 
particles. To this end let bx and by represent the displacements from 
the mean position (as, y) of a given particle. Our basic assumptions 
mean that bx, Sy and their derivatives are small quantities; it follows 
therefore that we may write 

dox 

— = u(x, y) = X = — m A cos ot cosh m(y + h) sin mx 
dt 

ddy 

— — = v(x, y) = y = mA cos ot sinh m(y + h) cos mx 

within the accuracy of our basic approximation. The constant A is 
an arbitrary factor fixing the amplitude of the wave. Hence we 
have upon integration 



(3.1.20) 



mA . . - . 

ox = — sin ot cosh m(y + h) sin mx, 

G 

. mA . . _ 

by = sin ot sinh m(y + h) cos mx. 



The motion of each particle takes place in a straight line the direction 
of which varies from vertical under the wave crests (cosmx = 1) to 
horizontal under the nodes (cos mx = 0). The motion also naturally 
becomes purely horizontal on approaching the bottom y = — h. 
These consequences of the theory are verified in practice, as indicated 
in Fig. 3.1.1, taken from a paper by Ruellan and Wallet (cf. [R.12]). 
The photograph at the bottom makes the particle trajectories visible in 
a standing wave; this is the final specimen in a series of photographs of 
particle trajectories for a range of cases beginning with a pure pro- 
gressing wave (cf. sec. 3.2), and continuing with superpositions of pro- 
gressing waves traveling in opposite directions and having the same 
wave length but not the same amplitudes, finally ending with a 
standing wave when the wave amplitudes of the two trains are equal. 
We proceed next to study the special class of three-dimensional 
standing waves that are simple harmonic in the time, and which 
depend only on the distance r from the y-axis. In other words, we 
seek standing waves having cylindrical symmetry. Again we seek 
solutions of (3.1.6) which satisfy (3.1.7). Only the case of water of 
infinite depth will be treated here, and hence (3.1.8) is replaced by 



42 



WATER WAVES 








Fig. 3.1.1. Particle trajectories in progressing and standing waves 

the condition that the solutions be bounded at oo in the negative 
//-direction as well as in the x- and z-directions. It is once more of 
interest to derive all possible standing wave solutions which are 
everywhere regular and bounded at oo because of the fact that the 
solutions in the present case behave quite differently from those 
obtained above for motions that are independent of the ^-coordinate. 



SIMPLE HARMONIC OSCILLATIONS 43 

In particular, we shall see that all bounded standing waves with 
cylindrical symmetry die out at oo like the inverse square root of 
the distance, while in two dimensions we have seen that the assump- 
tion that the wave amplitude dies out at oo leads to waves of zero 
amplitude everywhere. 

It is natural to make use of cylindrical coordinates in deriving 
our uniqueness theorem. Thus we write (3.1.6) in the form 



1 d / dq>\ d 2 cp 
r dr \ dr J dy 



(3.1.21) [r -I-) + --L = 0, ^y > - oo, ^ r < oo 



with r the distance from the ?/-axis. The assumption that cp depends 
only upon r and y and not on the angle 6 has already been used. 
For our purposes it is useful to introduce a new independent variable 
g replacing r by means of the relation 

(3.1.22) g = logr, 
in terms of which (3.1.21) becomes 

d 2 cp d 2 w 

(3.1.23) e- 2 * _r+_r=0, i/<0, -oo<e<oo. 

This equation holds, we observe, in the half-plane y < of the 
y, £-plane. The boundary condition to be satisfied at y = is (cf. 

(3.1.7)): 

(3.1.24) <p y — my = 0, m = o 2 /g. 

We wish to find all regular solutions of (3.1.23) satisfying (3.1.24) 
for which cp and <p y are bounded at oo. To this end we proceed along 
much the same lines as above (cf. (3.1.14) and the reasoning imme- 
diately following it) for the case of two dimensions, and introduce 
the function ip(q, y) by the identity 

(3.1.25) ip = (p y — mcp, y < 0, — oo < £ < oo. 

Since %p involves only a derivative of cp with respect to y and not 
with respect to q it follows at once that ip is a solution of (3.1.23). 
Since ip vanishes at y = from (3.1.24) it follows easily that it can 
be continued analytically into the upper half-plane y > by setting 
V(?» V) = — V>(q> ~y) aR d that the resulting function will be a 
solution of (3.1.23) in the entire q, y-plane. The function xp thus 
obtained will be bounded in the entire plane, since it was bounded 
in the lower half plane by virtue of the boundedness assump- 
tions with respect to cp. A theorem of S. Bernstein now yields 



44 WATER WAVES 

the result that ip is everywhere constant* if it is a uniformly bounded 
solution of (3.1.23) in the entire g, ?/-plane. Since ip vanishes on the 
y-axis it follows that ip vanishes identically. Consequently we con- 
clude from (3.1.25) that cp satisfies the relation 

(3.1.26) <p y — my = 0, y < 0. 

The most general function cp(g, y) satisfying this equation is 

(3.1.27) cp = e m yf(g) = e m vf(\ogr) = e m vg(r) 

with g(r) an arbitrary function. But op (r, y) is also a solution of 
(3.1.21) and hence g(r) is a solution of the ordinary differential 
equation 

(3.1.28) - — \r — \ +?n 



1 ±(r d l\ 
r dr \ dr] 



or, in other words, g(r) is a Bessel function of order zero: 

(3.1.29) g(r) = AJ (mr) + BY (mr). 

Since we restricted ourselves to bounded solutions only it follows 
that all solutions <p(r, y) of our problem are given by 

(3.1.30) g(r, y) = Ae m yJ (mr), m = o 2 /g, 

with A an arbitrary constant. Upon reintroduction of the time 
factor we have, therefore, as the only bounded velocity potentials 
the functions 

(3.1.31) 0(r, y; t) = Ae iat e m vJ (mr). 

As is well known, these functions behave for large values of r as 
follows : 

(3.1.32) 0(r, y; t) ~ Ae iat e my • [/ cos \mr — - 1 

' nmr \ 4/ 

and thus they die out like 1/\A% as stated above. 

In two dimensions we were able to find bounded standing waves 
of arbitrary phase (in the space variable) at oo. In the present case 
of circular waves we have found bounded waves with only one phase 
at oo. However, if we were to permit a logarithmic singularity at the 

* What is needed is evidently a generalization of Liouville's theorem to the 
elliptic equation (3.1.23) which has a variable coefficient. The theorem of Bern- 
stein referred to is much more general than is required for this special purpose, 
but it is also not entirely easy to prove (cf. E. Hopf [H.17] for a proof of it). 



SIMPLE HARMONIC OSCILLATIONS 45 

axis r = and thus admit the singular Bessel function Y (mr) as 
a solution of (3.1.28), we would have as possible velocity potentials 
the functions 

(3.1.33) 0(r, y; t) = Be iat e my Y (mr) 
which behave for large r as follows: 

(3.1.34) 0{r, y; t) d Be iot e my ]/ sin [mr — - ). 

" nmr \ 4/ 

Admitting solutions with a logarithmic singularity on the ?/-axis 
thus leads to standing waves which behave at oo in the same way 
as those which are everywhere bounded, except that they differ by 
90° in phase at oo. Thus waves having an arbitrary phase at oo can 
be constructed, but not without allowing a singularity. It has, however 
not been shown that (3.1.31) and (3.1.33) yield all solutions with this 
property. 

3.2. Simple harmonic progressing waves 

Since our boundary problem is linear and homogeneous we can 
reintroduce the time factors cos at and sin at and take appropriate 
linear combinations of the standing waves (3.1.5) to obtain simple 
harmonic progressing wave solutions in water of uniform depth of 
the form 

(3.2.1) = A cosh m(y + h) cos (mx ± at + a) 
with m and a satisfying 

(3.2.2) a 2 = gmtanhmh, 
as before. 

The wave, or phase, speed c is of course given by 

(3.2.3) c = er/ra, 

or, in terms of the wave length X = 2jr/m by 



(3.2.3^ c = [/°-tanh 

2tz X 



l/gA 
f ' 






It is useful to write the relation (3.2.2) in terms of the wave length 
X = 27z/m and then expand the function tanh mh in a power series 
to obtain 



(3.2.4) • a 2 = ^* 

X 



'2jih 1 /2ttM 3 



1 /2jthV 

3 VT/ 



46 WATER WAVES 

We see therefore that 

(2ji\ 2 fa 

— ) gh = m 2 gh as - -> 0, 

and hence that 

(3.2.6) cc^ Vgh if h\X is small. 

This last relation embodies the important fact that the wave speed 
becomes independent of the wave length when the depth is small compared 
with the wave length, but varies as the square root of the depth. This 
fact is in accord with what resulted in Chapter 2 upon linearizing 
the shallow water theory (cf. (2.2.16)) and the sentence immediately 
following), which led to the linear wave equation and to c = Vgh 
as the propagation speed for disturbances. We can gain at least a 
rough idea of the limits of accuracy of the linear shallow water 
theory by comparing the values of c given by c 2 = gh with those 
given by the exact formula 

(3.2.7) c^^tanh 2 ^ 

for various values of the ratio hjX. One finds that c as given by 
Vgh is in error by about 6 % if the wave length is ten times the 
depth and by less than 2 % if the wave length is twenty times the 
depth. The error of course increases or decreases with increase or 
decrease in h\X. 

In water of infinite depth, on the other hand, we have already 
observed (cf. (3.2.2)) that 

(3.2.8) c 2 = gXj2n. 

Actually, the error in c as computed by the formula c 2 = gXj2ji is 
already less than 1/2 % if hjX > J. One might therefore feel justified 
in concluding that variations in the bottom elevation will have but 
slight effect on a progressing wave provided that they do not result 
in depths which are less than half of the wave length, and observations 
seem to bear this out. In other words, the wave would not "feel" the 
bottom until the depth becomes less than about half a wave length. 
It is of interest to determine the paths of the individual water 
particles as the result of the passage of a progressing wave. As in the 
preceding section we take dx and dy to represent the displacements 
of a particle from its average position, and determine those dis- 
placements from the equations 



SIMPLE HARMONIC OSCILLATIONS 47 

door 

— = X = — Am cosh m(y + h) sin (mx + ot + a), 
eft 

ddy 

—— = = Am sinh m(?/ + ^) cos (m# + at -fa), 
a/ 

since is given by (3.2.1) in the present case. Integration of these 
equations yields 



(3.2.9) 



dx = cosh m(y + h) cos (mx -f- at + a), 

a 

j4 771 

by = sinh m(y -f /i) sin (m« + at + a), 



so that the path of a particle at depth y is an ellipse 

dx 2 dy 2 _ i 

6J 2 6 2 

with semi-axes a and 6 given by 

Am . . 
a = cosh ?n(?/ + /i) 

(7 

o = sinh ra(i/ + ^)- 

a 

On the bottom, y = — h, the ellipse degenerates into a horizontal 
straight line, as one would expect. Both axes of the ellipse shorten 
with increase in the depth. For experimental verification of these 
results, the discussion with reference to Fig. 3.1.1 should be con- 
sulted. In water of infinite depth the particle paths would be circles, 
as one can readily verify. The fact that the displacement of the 
particles dies out exponentially in the depth explains why a submarine 
need only submerge a slight distance below the surface — a half wave 
length, say — in order to remain practically unaffected even by severe 
storms. 



3.3. Energy transmission for simple harmonic waves of small 
amplitude 

In Chapter 1 the general formulas for the energy E stored in a 
fluid and its flux or rate of transfer F across given surfaces were 
derived for the most general types of motion. In this section 



48 WATER WAVES 

we apply these formulas to the special motions considered in the 
present chapter, that is, under the assumption that the free surface 
conditions are linearized. The formula for the energy E stored in a 
region R is (cf. (1.6.1)): 

(3.3.1 ) E = Q JJJ[i(^ 2 + 0J + 0!) + gy]dxdydz; 

R 

while the flux of energy F in a time T across a surface S G fixed in 
space is given by (cf. (1.6.5)): 

(3.3.2) F = ef +T (\\ t B £ dS ) dL 

s G 
We suppose first that the motion considered is the superposition 
of two standing waves which are simple harmonic in the time, as 
follows : 

(3.3.3) 0(x, y, z; t) = (p^x, y, z) cos at + (p^(x, y, z) sin at. 

Insertion of this in (3.3.2) with T — 2ji/a, i.e. for a time interval 
equal to the period of the oscillation, leads at once to the following 
expression for the energy flux F through S G : 

(3.3.4) F = Q ,jj( [ ^- (Pl d ^yS. 

s G 
One observes that the energy flux over a period is zero if either q) x 
or 9? 2 vanishes, i.e. if the motion is a standing wave: a fact which 
is not surprising since one expects an actual transport of energy 
only if the motion has the character of a progressing wave. Still 
another fact can be verified from (3.3.4) in our present cases, in which 
(f) x and q> 2 are, as we know, harmonic functions: if S G is a fixed closed 
surface in the fluid enclosing a region R Green's formula states that 

Jj(^ 2 ^ ~ q>i d -~j dS = JJJ^VVi ~ ?i V V 2 ) d x d y dz 

S G R 

provided that (p 1 and cp 2 have no singularities — sources or sinks for 
example — in R. In this case the energy flux F clearly vanishes since 
<p ± and 9? 2 are harmonic. Also one sees by a similar reasoning that the 
flux F over a period remains constant if S G is deformed without 
passing over singularities. In particular, the energy flux through a 
vertical plane passing from the bottom to the free surface of the water 



SIMPLE HARMONIC OSCILLATIONS 49 

in a two-dimensional motion would be the same (per unit width of 
the plane) for all positions of the plane provided that no singularities 
are passed over. This fact makes it possible, if one wishes, to con- 
sider the energy in the fluid as though the energy itself were an 
incompressible fluid, and to speak of its rate of flow. 

In the literature dealing with waves in all sorts of media, but 
particularly in dispersive media, it is indeed commonly the custom 
to introduce the notion of the velocity of the flow of energy ac- 
companying a progressing wave, and to bring this velocity in relation 
to the kinematic notion of the group velocity (to be discussed in 
the next section). The author has found it difficult to reconcile him- 
self to these discussions, and feels that it would be better to discard 
the difficult concept of the velocity of transmission of energy, since 
this notion is not of primary importance, and nothing can be ac- 
complished by its use which cannot be done just as well by using 
the well-founded and clear-cut concept of the flux of energy through 
a given surface. On the other hand, the notion is used in the literature 
(and probably will continue to be used) and consequently a dis- 
cussion of it is included here, following pretty much the derivation 
given by Rayleigh in an appendix to the first volume of his Sound 
[R.4]. In the next section, where the notion of group velocity is 
introduced, some further comments about the concept of the velocity 
of transmission of energy will be made. 

We consider the energy flux per unit breadth across a vertical 
plane in the case of a simple harmonic progressing wave in water 
of uniform depth (or, in view of the above remarks, across any surface 
of unit breadth extending from the bottom to the free surface). The 
velocity potential is given by (cf. (3.2.1)) 

(3.3.5) = A cosh m(y + h) cos (mx + at + a) 
and (3.3.2) yields 

(3.3.6) F = A 2 Q om C' +2n,a (P cosh 2 m(y + h)dy\ sin 2 (mx + at) dt 

for the flux across a strip of unit breadth in the time T = 2jz/a, 
the period of the oscillation. Hence the average flux per unit time 
is given by 

(3 3 7) F - - — A2 Q Gmh d i sinh 2mh \ 

T 4 \ 2mh J 

since the average of sin 2 6 over a period is 1/2. We have also taken 



50 WATER WAVES 

tj = o in the upper limit of the integral in (3.3.6) and thus neglected 
a term of higher order in the amplitude. It is useful to rewrite the 
formula (3.3.7) in the following form through use of the relations 
a 2 = gm tanh mh and c = ojm: 

(3.3.8) F av = A -^- cosh 2 mh • U, 

with U a quantity having the dimensions of a velocity and given 
by the relation 

1 / %™>h \ 

(3.3.9) U = -c 1 + . 

2 \ sinh 2mhJ 

Next we calculate the average energy stored in the water as a result 
of the wave motion with respect to the length in the direction of 
propagation of the wave. This is obtained from (3.3.1) by calculating 
first the energy E k over a wave length X = 2n\m at any arbitrary 
fixed time. In the present case we have 

Ex — E = m 2 Q I f H A 2 sinh 2 m(y -\-h) cos 2 (mx -j-ot +a) 

J —hJ 

(3.3.10) + \ A 2 cosh 2 m(y + h) sin 2 (mx + at + a)] dxdy 
+ e p£gydxd y , 

in which the constant E refers to the potential energy of the water 
of depth h when at rest. On evaluating the integrals, and ignoring 
certain terms of higher order, we obtain for the energy between two 
planes a wave length apart arising from the passage of the wave the 
expression 

(3.3.11) Ex- E = ^— X cosh 2 mh, 

as one finds without difficulty. Thus the average energy E av in the 
fluid per unit length in the ^-direction which results from the motion 
is given by 

(3.3.12) E av = A ^- cosh 2 mh. 

2g 

Upon comparison with equation (3.3.8) we observe that E av is 
exactly the coefficient of TJ in the formula (3.3.8) for the average 
energy flux per unit time across a vertical plane. It therefore follows, 
assuming that no energy is created or destroyed within the fluid 



SIMPLE HARMONIC OSCILLATIONS 51 

itself, that the energy is transmitted in the direction of propagation of 
the wave on the average with the velocity U. As we see from (3.3.9) 
the velocity U is not the same as the phase or propagation velocity c; 
in fact, U is always less than c: for water of infinite depth it has the 
value c/2 and it increases with decrease in depth, approaching the 
phase velocity c as the depth approaches zero. 

3.4. Group velocity. Dispersion 

In any body of water the motion of the water in* general consists 
of a superposition of waves of various amplitudes and wave lengths. 
For example, the motion of the water due to a disturbance over a 
restricted area of the surface can be analyzed in terms of the super- 
position of infinitely many simple harmonic wave trains of varying 
amplitude and wave length; such an analysis will in fact be carried 
out in Chapter 6. However, we know from our previous discussion 
(cf. (3.2.7)) that the propagation speed of a train of waves is an 
increasing function of the wave length — in other words, the wave 
phenomena with which we are concerned arc subject to dispersion — 
and thus one might expect that the waves would be sorted out as 
time goes on into various groups of waves such that each group 
would consist of waves having about the same wave length. We wish 
to study the properties of such groups of waves having approximately 
the same wave length. 

Suppose, for example, that the motion can be described by the 
superposition of two progressing waves given by 

(3 4 1) [0 1 = Asin (mx - at) 

[0 2 = A sin ([m + dni]x — [a + do]t) 

with dm and do considered to be small quantities. The superposition 
of the two wave trains yields 



= 2 A cos - (xdm — tdo) sin 
(3.4.2) 2 



dm 

m + — ■ 
2 



do 



= B sin (m'x — o't) 

with m' = m + dm/2, a' = a -f do/2. Since dm and do are small it 
follows that the function B varies slowly in both x and t so that 
is an amplitude-modulated sine curve at each instant of time, as 
indicated schematically in Figure 3.4.1. In addition, the ''groups" 
of waves thus defined — in other words the configuration represented 



52 WATER WAVES 

by the dashed curves of Figure 3.4.1.— advance with the velocity 
do J dm in the ^-direction. In our problem a will in general be a function 




Fig. 3.4.1. Wave groups 

of m so that the velocity U of the group is given approximately 
by da /dm, or, in terms of the wave length X = 27i/m and wave velocity 
c = a/m, by 

(3.4.3) U = *M = C - X *. 

dm dX 

The matter can also be approached in the following way (cf. 
Sommerfeld [S.13]), which comes closer to the more usual cir- 
cumstances. Instead of considering the superposition of only two 
progressing waves, consider rather the superposition, by means of 
an integral, of infinitely many waves with amplitudes and wave 
lengths which vary over a small range: 

(3.4.4) = ( m ° +£ A(m) exp {i(mx — at)} dm. 

J m -e 

The quantity mx — at can be written in the form 

(3.4.5) mx — at = m x — a t + (m — m )x — (a — a )t. 
From (3.4.4) one then finds 

(3.4.6) = C exp {i(m x — a t)}, 

in which the amplitude factor C is given by 

(3.4.7) C = \° E A(m) exp{i[(m — m )x — (a — a )t]} dm. 

J m -e 

We are interested here in seeking out those places and times (if any) 
where the function C represents a wave progressing with little change 
in form, since (3.4.6) will then furnish what we call a group of waves. 
Since x and t occur only in the exponential term in (3.4.7), it follows 
that the values of interest are those for which this term must be 
nearly constant, i.e. those for which (m — m )x — (a — a )tc^ const. 



SIMPLE HARMONIC OSCILLATIONS 53 

It follows that the propagation speed of such a group is given by 
{a — o )l(m — m ), and if (m — m ) is small enough we obtain again 
the formula (3.4.3). 

Evidently, it is important for this discussion of the notion of group 
velocity that the motion considered should consist of a superposition 
of waves differing only slightly in frequency and amplitude. In 
practice, the motions obtained in most cases —through use of the 
Fourier integral technique, for example,— are the result of super- 
position of waves whose frequencies vary from zero to infinity and 
whose amplitudes also vary widely. However, as we shall see in 
Chapter 6, it happens very frequently that the motion at certain 
places and times is approximated with good accuracy by integrals 
of the type given in (3.4.4) with s arbitrarily small. (This is, indeed, 
the sense of the principle of stationary phase, to be treated in Chap- 
ter 6.) In such cases, then, groups of waves do exist and the dis- 
cussion above is pertinent. 

In our problems the relation between wave speed and wave length 
is given by (3.2.2) and consequently the velocity U of a group is 
readily found, from (3.4.3), to be 

(3.4.8 U = -c (l + . 

2 \ sinh 2mh) 

We observe that the group velocity has the same value as was given 
in the preceding section for the average rate of propagation of energy 
in a uniform train of waves having the same wave length as those 
of the group. In other words, the rate at which energy is propagated 
is given by the group velocity and not the phase velocity. This is 
often considered as the salient fact with respect to the notion of 
group velocity. As indicated already in the preceding section, the 
author does not share this view, but feels rather that the kinematic 
concept of group velocity is of primary significance, while the notion 
of velocity of propagation of energy might better be discarded. It 
is true that the two velocities, in spite of the fact that one is derived 
from dynamics while the other is of purely kinematic origin, turn 
out to be the same — not only in this case, but in many others as 
well* — but it is also true that they are not always the same — for 
example, the two velocities are not the same if there is dissipation 
of energy in the medium. In addition, we have seen in the preceding 
section that the notion of velocity of energy can be derived when no 

* A general analysis of the reason for this has been given by Broer [B.18]. 



54 WATER WAVES 

wave group exists at all— we in fact derived this velocity for the case 
of a wave having but one harmonic component. 

In Chapter 6 we shall have occasion to see how illuminating the 
kinematic concept of a group and its velocity can be in interpreting 
and understanding the complicated unsteady wave motions which 
arise when local disturbances propagate into still water. 



CHAPTER 4 



Waves Maintained by Simple Harmonic Surface Pressure 
in Water of Uniform Depth. Forced Oscillations 



4.1. Introduction 

In our previous discussions we have considered always that the 
pressure at the free surface was constant (usually zero) in both space 
and time. In other words, only the free oscillations were treated and 
the problems were, correspondingly, linear and homogeneous boun- 
ary value problems. Here we wish to consider two problems in which 
the surface pressure p is simple harmonic in the time and the resulting 
motions are thus forced oscillations; the problems then also have a 
nonhomogeneous boundary condition. In the first such problem we 
assume that the motion is two-dimensional and that the surface pres- 
sure is a periodic function of the space coordinate x over the entire 
ir-axis; in the second problem the surface pressure is assumed to be 
zero except over a segment of finite length of the #-axis. In these 
problems the depth of the water is assumed to be everywhere infinite, 
but the corresponding problems in water of constant finite depth 
can be, and have been, solved by much the same methods. 

The formulation of the first two problems is as follows. A velocity 
potential &{x, y; t) is to be determined which is simple harmonic in 
the time / and satisfies 

(4.1.1) V 2 = for y < 0. 
The surface pressure p(x; t) is given by 

(4.1.2) p(x; t) = p(x) sin at, 

and the boundary conditions at the free surface are the dynamical 
condition (cf. (2.1.20^) 

1 

(4.1.3) r] = --0 t -plQg, 

o 

55 



50 WATER WAVES 

and the kinematic condition 

(4.1.4) rjt = ®v 

The last condition means that no kinematic constraint is imposed 
on the surface — it can deform freely subject to the given pressure 
distribution. In addition, we require that t and y should be 
uniformly bounded at oo. This means effectively that the vertical 
displacement and vertical velocity components are bounded. In 
section 4.3, the amplitude p(x) of the surface pressure p will have 
discontinuities at two points and we shall impose appropriate con- 
ditions on at these points when we consider this case. 

We seek the most general simple harmonic solutions of our problem; 
they have the form 

(4.1.5) — cp(x, y) cos at -f ip(x, y) sin at. 

The functions 92 and ip are of course harmonic in the lower half 
plane. The conditions (4.1.2), (4.1.3), and (4.1.4) are easily seen to 
yield for the function <p the boundary condition 

a _ 

(4.1.6) cp y — mcp = — — p(x) for y = 

with the constant m defined by 

(4.1.7) m = a 2 /g; 
while for ip they yield the condition 

(4.1.8) ip y — imp = for y = 0. 

The phase sin at assumed for p in (4.1.2) has the effect that ip satisfies 
the homogeneous free surface condition, as one sees. 

We know from the first section of the preceding chapter that the 
only bounded and regular harmonic functions \p which satisfy the 
condition (4.1.8) are given by 

{cos mx 

In the next two sections we shall determine the function cp(x, y), 
i.e. that part of which has the phase cos at, in accordance 
with two different choices for the amplitude p(x) of the surface 
pressure p. 



SIMPLE HARMONIC SURFACE PRESSURE 57 

4.2. The surface pressure is periodic for all values of x 

We consider now the case in which the surface pressure is periodic 
in x such that p(x) in (4.1.2) and (4.1.6) is given by 

(4.2.1 ) p(x) = P sin he, — oo < x < oo. 
One verifies at once that the following function cp(x, y): 

aP e Xv 

(4.2.2) <p(x, y) = r sin he 

qg m — A 

is a harmonic function which satisfies the free surface boundary 
condition (4.1.6) imposed in the present case. Since the difference % 
of two solutions q> lt cp 2 both satisfying all of our conditions would 
satisfy the homogeneous boundary condition % y — m% = 0, it follows 
that all solutions cp of our boundary value problem can be obtained by 
adding to the special solution given by (4.2.2) any solution of the 
homogeneous problem, and these latter solutions are the functions 
y) given by (4.1.9) since # satisfies the same conditions as tp. Therefore 
the most general simple harmonic solutions of the type (4.1.5) are 
given in the present case by 



(4.2.3) 0(x,y;t) 



aP e Xv [ cos mx 

sin Ix + Ae mv 



_gg m — A [sin mx 

(cos mx 1 
\ sin oU 



cos oi 



{cos mx 1 
sii 
sin mx J 



with A and B constants which are at our disposal. In other words, 
the resulting motions are, as usual in linear vibrating systems, a 
linear combination of the forced oscillation and the free oscillations. 
These solutions — without the uniqueness proof— seem to have been 
given first by Lamb [L.2]. 

We observe that the case X = m must be excluded, and that if A 
is near to m large amplitudes of the surface waves are to be expected. 
This means physically, as one sees immediately, that waves of large 
amplitude are created if the periodic surface pressure distribution 
has nearly the wave length which belongs to a surface wave of the 
same frequency for pressure zero at the surface — that is, the wave 
length of the corresponding free oscillation. 

If instead of (4.1.2) we take the surface pressure as a progressing 
wave of the form 

(4.2.4) p( X ; t) = H sin (at - he) 



58 WATER WAVES 

it is readily found that progressing surface waves result which are 
given by 

Ho e Xv 

(4.2.5) 0(x, y;t) = r cos (at — he). 

gg m — A 

To this one may, of course, add any of the wave solutions which 
occur under zero surface pressure. Again one observes an odd kind 
of "resonance" phenomenon: large amplitudes are conditioned by 
the wave length in space of the applied pressure once the frequency 
has been fixed. 

4.3. The variable surface pressure is confined to a segment of the 
surface 

In this section we consider the case in which the surface pressure p 

{P sin ot, I x I < a 
i.i>.' y = 

with P a constant. Some of the motions which can arise under such 
circumstances are discussed by Lamb [L.2] in the paper quoted above. 
However, here as elsewhere, Lamb assumes fictitious damping 
forces* in order to be rid of the free oscillations and thus achieve a 
unique solution, and he also makes use of the Fourier integral tech- 
nique which we prefer to replace by a different procedure. In fact, 
the present problem is a key problem for this Part II and a peg upon 
which a variety of observations important for other discussions in later 
chapters will be hung. As we shall see, the present problem is also 
decidedly interesting for its own sake, although Lamb strangely 
enough made no attempt in his paper to point out the really striking 
results. 

In addition to prescribing the pressure p through (4.3.1) it is 
necessary to add to the conditions imposed in section 4.1 appropriate 
conditions at the points (^ a, 0) where p has discontinuities. In 
view of (4.1.3) it is clear that a finite discontinuity in t or r\ or 
both must be admitted and it seems also likely that the derivatives 
X and y of would be unbounded near these points. We shall make 

* Lamb assumes resistances which are proportional to the velocity. In this 
way the irrotational character of the flow is preserved, but it is difficult to see 
how such resistances can be justified mechanically. It would seem preferable 
to secure the uniqueness of the solution in unbounded domains by imposing 
physically reasonable conditions on the behavior of the waves at infinity. 



SIMPLE HARMONIC SURFACE PRESSURE 59 

the following requirements 

(4.3.2) t bounded; y = 0(g~ 1+e ), e > 

in a neighborhood of the points (^ a, 0) with g the distance from 
these points. This means, in particular, that the surface elevation is 
bounded near these points and that the singularity admitted is not 
as strong as that of a source or sink. We recall that @ t and y were 
required to be uniformly bounded at oo. 

The stipulations made so far do not ensure the uniqueness of the 
solution of our problem any more than similar conditions ensured 
uniqueness of the solution of the problem treated in the preceding 
section. However, we have in mind now a physical situation in which 
we expect the solution to be unique: We imagine the motion resulting 
from the applied surface pressure p given by (4.3.1) to be the limit 
approached after a long time subsequent to the application of p to 
the water when initially at rest. Under these circumstances one feels 
instinctively that the motion of the water far away from the source 
of the disturbance should have the character of a progressing wave 
moving away from the source of the disturbance, since at no time 
is there any reason why waves should initiate at infinity. (We shall 
show (cf. (6.7)) that the motion of the water arising from such initial 
conditions actually does approach, as the time increases without 
limit, the motion to be obtained here.) Consequently we add to our 
conditions on the condition — often called the Sommerfeld condition 
in problems concerning electromagnetic wave propagation— that the 
waves should behave at oo like progressing waves moving away from 
the source of the disturbance. As we shall see, this qualitative condition 
leads to a unique solution of our problem. 

In solving our problem there are some advantages to be gained by 
not stipulating at the outset that the Sommerfeld condition should 
be satisfied, but to obtain first all possible solutions of the form 
(4.1.5), and only afterwards impose the condition. We have therefore 
to find the harmonic functions cp which satisfy the condition (cf. 
(4.1.6) and (4.3.1)) 

{ c, I x I < a 

(4.3.3) 9-.-*9» = ( 0j |0| ; a . = 

with 

(4.3.4) m = a 2 jg, c = — — 

eg 



60 WATER WAVES 

on the free surface, and the boundedness conditions which follow 
from those imposed on 0: 

I (p and (p y bounded at oo, 

\ <p bounded and cp v = 0(o~ 1+e ), s > 0, at x = ± a. 

The functions \p in (4.1.5), i.e. those which yield the waves of phase 
sin at in 0, satisfy the same conditions as in section 4.1 and are 
therefore given by (4.1.9). We have therefore only to determine the 
functions (p. To this end it is convenient to introduce new dimen- 
sionless quantities 

(4.3.6) x x = mat, y x = my, a x = ma 

together with c 1 = c/m so that the free surface condition (4.3.3) 
takes the form 

I A orrs l € l> 1*1 i =^1 

(4 - 8 - 7) ^~Ho, i«m >v yi = 

In what follows we use the condition in this form but drop the sub- 
scripts for the sake of convenience. 

In most of the two-dimensional problems treated in the remainder 
of Part II we make use of the fact that any harmonic function 
(p(x, y) can be taken as the real part of an analytic function f(z) of 
the complex variable z = x -{- iy and write 

(4.3.8) /(») = <p(x, y) + iy(x, y) = f(x + iy). 

In our present problem f(z) is defined and analytic in the lower half 
plane. To express the surface condition (4.3.7) in terms of f(z) we 
write 

fy - 9 = (I, - i)v = *«(^ - 1) W +iy) = *'(^~ ^/W 

in which the symbol £%e means that the real part of what follows is 
to be taken. Consequently the free surface condition has the form: 

(4.3.9) <Pv - (p = ^(ir-n = [l J; 1 !;, y = o. 






SIMPLE HARMONIC SURFACE PRESSURE 61 

We now introduce a new analytic function F(z) by the equation* 

(4.3.10) F(z) = if'(z) - f(z) 

and seek to determine F(z) uniquely through the conditions imposed 
on (p = 0te f{z). We observe to begin with that F(z) satisfies the 
condition 



!:. 



' I | > a ' y = °' 



in view of (4.3.9). We show now that F(z) is uniquely determined 
within an additive pure imaginary constant, as follows: Suppose 
that G(z) = F x (z) — F 2 (z) is the difference of two functions F(z) 
satisfying the conditions resulting from (4.3.10) through those on 
f(z). Then 0te G(z) would vanish on the entire real axis, except 
possibly at x = i a > as one sees from (4.3.11). Hence 0te G(z) is a 
potential function which can be continued analytically by reflection 
over the real axis into the entire upper half plane; it will then be 
defined and single-valued in the whole plane except for the points 
(i a, 0). At oo, 0te G(z) is bounded in the lower half plane, while 
&e G(z) = 0(g~ 1+e ), s > 0, at x = ± a m view of the regularity 
conditions and the definition of G(z). These boundedness properties 
are evidently preserved in the analytic continuation into the upper 
half plane. Consequently &e G(z) has a removable singularity at the 
points x = ± a on the real axis since the singularity is weaker than 
a pole of first order and the function is single-valued in the neigh- 
borhood of these points. Thus 0te G(z) is a potential function which 
is regular and bounded in the entire plane, and is zero on the real 
axis; by Liouville's theorem it is therefore zero everywhere. Con- 
sequently the analytic function G(z) is a pure imaginary constant, 
and the result we want is obtained. On the other hand it is rather 
easy to find a function F(z) which has the prescribed properties — for 
example by first finding its real part from (4.3.11) through use of 
the Poisson integral formula. We simply give it: 

ic z — a 
(4.3.12) F(z) = - log — — ; 

7i z + a 

one verifies readily that it has all of the required properties. We 
take that branch of the logarithm which is real for (z — a)/(z + a) 

* This device has been used by Kotschin [K.14], and it was exploited by Lewy 
[L.8] and the author [S.18] in studying waves on sloping beaches. 



62 



WATER WAVES 



real and positive. 

Once F(z) has been uniquely determined, the complex velocity 
potential f(z) is restricted to the solutions of the first order ordinary 
differential equation (4.3.10), which means that the solutions depend 
only on the arbitrary constant which multiplies the non-vanishing 
solution e~ iz of the homogeneous equation if'(z) — f = 0. But 
0te {A + iB)e~ iz = e y (A cos x + B sin x) and these are the standing 
wave solutions for the case of surface pressure p = 0. The most 
general solution of (4.3.10), with F(z) given by (4.3.12), can be 
written, as one can readily verify, in the form 



(4.3.13) 



f(z)=-e- 

71 



t — a 

,tt i g _ — . — ^ 



t + a 

with the initial point z and the path of integration any arbitrary 
path in the slit plane. Changing z obviously would have the effect 
of changing the additive solution of the homogeneous equation. It 
is convenient to replace (4.3.13) by the following expression, obtained 
through an integration by parts: 

e it | _ 1 

J +zoo \t — a t + a] 



(4.3.14) /(*) 



log 



z + a 



dt 



and at the same time to fix the path of integration as indicated in 



t- plane 






X- plane 






I ~' a 


+a 1/ 




/z 




-*t 



(a) (b) 

Fig. 4.3.1a,b. Path of integration in /-plane 

Figure 4.3.1. This path comes from oo along the positive imaginary 
axis and encircles the origin, leaving it and the point (—a, 0) to 



SIMPLE HARMONIC SURFACE PRESSURE 



63 



the left. Use has been made of the fact that log (z — a) / (z -\- a) ^0 
when z -» oo; we observe also that the integrals converge on account 
of the exponential factor. 

That q?(x, y) = 2%e f(z) as given through (4.3.14) satisfies the 
boundary conditions imposed at the free surface and the regularity 
condition at the points (^ a, 0) is easy to verify. We proceed to 
discuss the behavior of f(z) at oo (always for z in the lower half plane). 
For this purpose it suffices to discuss the integrals 



I(z) 



:oo t =b a 



dt since the function log 



Iz-aX 

\z+a) 



behaves like 1/z at oo (as one readily sees). To this end we integrate 
once by parts to obtain 

,i(t-z) 



/(«) 



z i a 



j; 



(t±ay 



dt. 



We suppose that the curved part of the path of integration in Figure 
4.3.1a is an arc of a circle. It follows at once that the complex number 
t — z has a positive imaginary part on the path of integration as 
long as the real part of z is negative, and hence we have 



I /(«) I ^ 



1 



z ± a 



+ 



/: 



| dt 



t±a\' 



< 



z± a 



+ 



I 



\dt\ 



t±a 



Consequently I(z ) behaves like 1/z at infinity when the real part of 
z is negative, and f(z) likewise. The situation is different, however, 
if the real part of z is positive. To study this case, we add and subtract 
circular arcs, as indicated in Figure 4.3.1b, in order to have an 
integral over the entire circle enclosing the singularities at ±« 
as well as over a path symmetrical to the path in Figure 4.3.1a. 
By the same argument as above, the contribution of the integral 
over the latter path behaves like 1/z at oo, and hence the non- 
vanishing contribution arises as a sum of residues at the points ± a. 
These contributions to I(z) are at once seen to have the values 
2me~ lz e Tia . Thus we may describe the behavior of f(z) as given 
by (4.3.14) at oo as follows: 



(4.3.15) f(z) - 



— 4ci (sin a)e~ iz + O ( — j 



for 0te z < 
for &e z > 0. 



64 WATER WAVES 

From (4.3.10) and (4.3.12) one sees that f'(z) has the same behavior 
at oo as f(z), except for a factor — i. Hence f(z), and with it 
y(x, y) = 0te f(z), has the postulated behavior at oo. It is convenient 
to write down explicitly the behavior of <p(x, y) at oo: 



(4.3.16) tp(x, y) =0tej{z\ 



»(t) 

•+°(t) 



for x < 0, 
4c sin a e y sin x + I — I for x > 0. 



It follows that all simple harmonic solutions of our problem are 
given by linear combinations of 

(4.3.17) 0{x, y; t) = (0te f(z) + Ae y sin x + Be y cos x) cos ot 
and 

(4.3.18) 0(x, y; t) = (Ce y sin x + De y cos x) sin ot 

in which A, B, C, and D are arbitrary constants, and f(z) is given 
by (4.3.14). In other words, the standing waves cp(x, y) cos ot just 
found above, together with the standing waves which exist for 
vanishing free surface pressure, constitute all possible standing waves. 
We now impose the condition that the wave @(x, y; t) we seek 
behaves like an outgoing progressing wave at oo, i.e. that it behaves 
like 

£_: e y (H sin (x + ot) + K cos (x + ot)) at x = — oo 
and like 

S + : e y (L sin (x — ot) + M cos (x — ot)) at x = + oo. 

In view of the behavior of 99^, ?/) = 0te f(z) at ^r = — 00 (cf. (4.3.16)), 
i.e. the fact that it dies out there, it is clear that we may combine 
the standing wave solutions (4.3.17) and (4.3.18) in such a way as 
to obtain a progressing wave solution 

(4.3.19) 0(x, y; t) = e y (H sin (x + at) + K cos (x + at)) 

+ (p(x, y) cos at 

valid everywhere and which satisfies the condition S_, with the two 
constants H and K still arbitrary. At x = + 00 this wave has the 
behavior 

(4.3.20) 0(x, y; t) = e y [{H sin (x -f at) +A"cos (x + at)) 

— 4c sin a sin a; cos at] + O I — I 



SIMPLE HARMONIC SURFACE PRESSURE 65 

in view of (4.3.16). In order that S+ should hold for this solution 
for all t one sees readily that the constants H and K must satisfy 
the linear equations 

f L = H — 4c sin a 
(4.3.21) [L = - H 

[M =K, M = -K, 

from which we conclude that 

L = — 2c sin a, H = 2c sin a 



(4.3.22) 

■ M = K = 0. 

Thus the solution is now uniquely determined through imposition 
of the Sommerfeld condition, and can be expressed as follows: 

2P ° / 1 \ 

(4.3.23) 0{x, y;t) = sin ma e™y sin (mx — at) + O — , x > 

ggm \ r J 

upon reintroduction of the original variables and parameters (cf. 
(4.3.6)), with 0(1 jr) representing a function which dies out at 
infinity like 1/r. The function of course yields a wave with sym- 
metrical properties with respect to the ?/-axis. We observe that 
the wave length X = 27i\m of these waves at oo is the same as that of 
free oscillations of the same frequency, as one would expect. 

The most striking thing about the solution is the fact that for 
certain frequencies and certain lengths of the segment over which 
the periodic pressure differs from zero, the amplitude of the progressing 
wave is zero at oo; this occurs obviously for sin ma = 0, i.e. for 
ma = kjz, k = 1, 2, 3, . . .. Since m = 2tzjX, with X the wave length 
of a free oscillation of frequency a, it follows that the amplitude of 
the progressing wave at oo vanishes when 

(4.3.24) 2a = kX, k = 1, 2, . . ., 

i.e. when the length of the segment on which the pressure is applied 
is an integral multiple of the wave length of the free oscillation having 
the same frequency as the periodic pressure. This does not of course 
mean that the entire disturbance vanishes, but only that the motion 
in this case is a standing wave given by 

(4.3.25) 0(x, y; t) = cp(x, y) cos gU 

since the quantities H and K in (4.3.19) are now both zero. Since 
<p now behaves like 1/r at both infinities, the amplitude of the standing 



66 WATER WAVES 

wave tends to zero at infinity. A wave generating device based on the 
physical situation considered here would thus be ineffective at certain 
frequencies. It is clear that no energy is carried off to infinity in 
this case, and hence that the surface pressure p on the segment 
— a ^ x 5^ + a can do no net work on the water on the average. 
Since rj t = @ y it follows that the rate at which work is done by the 
pressure p (per unit width at right angles to the x, ?/-plane) is 

pcp y cos at dx, and since p has the phase sin at it is indeed clear 

J — o 

that the average rate of doing work is zero in this case. 

There is a limit case of the present problem which has considerable 
interest for us. It is the limit case in which the length of the segment 
over which p is applied shrinks to zero while the amplitude P of p 
increases without limit in such a way that the product 2aP approaches 
a finite limit. In this way we obtain the solution for an oscillating 
pressure point. One sees easily that the function f(z) given by (4.3.13), 
which yields the forced oscillation in our problem, takes the following 
form in the limit: 



C C z e u 

— e -iz 

n J t 



(4.3.26) f(z) = —e~ iz — dt, 

n J t 

with C the real constant 2aPo/gg. At oo this function behaves as 
follows 



(4.3.27) f(z) 



O ( — J for 0te z < 0, 

2Ci e~ iz + O I — | for &ez>0. 



In this limit case of an oscillating pressure point we see that there are 
no exceptional frequencies: application of the external force always 
leads to transmission of energy through progressing waves at oo. 
The singularity of f(z) at the origin is clearly a logarithmic singularity 
since f(z) behaves near the origin like 

C C z dt 

(4.3.28) f(z) = -e~ iz —+.... 

n J t 

We see that a logarithmic singularity is appropriate at a source or 
sink of energy when the motion is periodic in the time. 



SIMPLE HARMONIC SURFACE PRESSURE 



67 



4.4. Periodic progressing waves against a vertical clift 

With the aid of the complex velocity potential defined by (4.3.13) 
we can discuss a problem which is different from the one treated in 
the preceding section. The problem in question is that of the deter- 
mination of two-dimensional progressing waves moving toward a 
vertical cliff, as indicated in Figure 4.4.1. The cliff is the vertical 



iSft 






11 



x = a 



s$&s 



Fig. 4.4.1. Waves against a vertical cliff 



plane containing the y-axis. As in the preceding section, we assume 
also that a periodic pressure (cf. (4.3.1)) is applied over the segment 
5^ x ^ a at the free surface. To solve the problem we need only 
combine the standing waves given by (4.3.17) and (4.3.18) in such 
a way as to obtain progressing waves which move inward from the 
two infinities, and this can be done in the same way as in section 4.3. 
The result will be again a wave symmetrical with respect to the 
t/-axis, and hence one for which X = along the ?/-axis; thus such 
a wave satisfies the boundary condition appropriate to the vertical 
cliff. We would find for the velocity potential the expression, valid 
for x > 0: 

2Po / 1 \ 

(4.4.1 ) 0(x, y; t) = sin ma ^Tsin (mx + at)] + O — 

Qgm \rj 

with 0(1 /r) a function behaving like 1/r at oo but with a singularity 
at (a, 0). In order to obtain a system of waves which are not reflected 
back to oo by the vertical cliff it was necessary to employ a mechanism 
—the oscillating pressure over the segment ^ « ^ a on the free 



68 WATER WAVES 

surface— which absorbs the energy brought toward shore by the in 
coming wave. However, the particular mechanism chosen here, i.e. 
one involving an oscillatory pressure having the same frequency as 
the wave, will not always serve the purpose since the amplitude A 
of the surface elevation of the progressing wave at oo is given, from 

(4.4.1) and (4.1.4), by 

2P 

(4.4.2) A = — sin ma. 

Thus the ratio of the pressure amplitude P applied on the water 
surface near shore to the amplitude of the wave at oo would obviously 
become oo when sin ma = 0. In other words, such a mechanism would 
achieve its purpose for waves whose wave length X at oo satisfies 
the relation a = k A/2, with k any integer, only if infinite pressure 
fluctuations at the shore occur. Presumably this should be interpreted 
as meaning that for these wave lengths the mechanism at shore is 
not capable of absorbing all of the incoming energy, or in other words, 
some reflection back to oo would occur. This remark has a certain 
practical aspect: a device to obtain power from waves coming toward 
a shore based on the mechanism considered here would function 
differently at different wave lengths. 

It is of interest in the present connection to consider the same limit 
case as was discussed at the end of the preceding section, in which 
the segment of length a shrinks to zero while Pa remains finite. In 
this case no exceptional wave lengths or frequencies occur. However, 
the limit complex potential now has a logarithmic singularity at the 
shore line, as we noticed in the preceding section, and the amplitude 
of the surface would therefore also be infinite at the shore line. What 
would really happen, of course, is that the waves would break along 
the shore line if no reflection of wave energy back to oo occurred, 
and the infinite amplitude obtained with our theory represents the 
best approximation to such an essentially nonlinear phenomenon 
that the linear theory can furnish. 

This limit case represents the simplest special case of the problem 
of progressing waves over uniformly sloping beaches which will be 
treated more generally in the next chapter. However, the present 
case has furnished one important insight: a singularity of the complex 
velocity potential is to be expected at the shore line if the condition 
at oo forbids reflection of the waves back to oo, and the singularity 
should be at least logarithmic in character. 



CHAPTER 5 



Waves on Sloping Beaches and Past Obstacles 

5.1. Introduction and summary 

Perhaps the most striking — and perhaps also the most fascinating— 
single occurrence among all water wave phenomena encountered in 
nature is the breaking of ocean waves on a gently sloping beach. 
The purpose of the present chapter is to analyze mathematically the 
behavior of progressing waves over a uniformly sloping beach insofar 
as that is possible within the accuracy of the linearized theory for 
waves of small amplitude; that is, within the accuracy of the theory 
with which we are concerned in the present Part II. Later, in Chapter 
10.10, we shall discuss the breaking of waves from the point of view 
of the nonlinear shallow water theory. 

To begin with, it is well to recall the main features of what is often 
observed on almost any ocean beach in not too stormy weather. 
Some distance out from the shore line a train of nearly uniform 
progressing waves exists having wave lengths of the order of say 
fifty to several hundred feet. These waves can be considered as simple 
sine or cosine waves of small amplitude. As the waves move toward 
shore, the line of the wave crests and troughs becomes more and 
more nearly parallel to the shore line (no matter whether this was 
the case in deep water or not), and the distance between successive 
wave crests shortens slightly. At the same time the height of the 
waves increases somewhat and their shape deviates more and more 
from that given by a sine or cosine— in fact the water in the vicinity 
of the crests tends to steepen and in the troughs to flatten out until 
finally the front of the wave becomes nearly vertical and eventually 
the water curls over at the crest and the wave breaks. These ob- 
servations are all clearly borne out in Figures 5.1.1, 5.1.2, which 
are photographs, given to the author by Walter Munk of the Scripps 
Institution of Oceanography, of waves on actual beaches. At the 
same time, it should be stated here that the breaking of waves also 
occurs in a manner different from this— a fact which will be discussed 

69 




Fig. 5.1.1. Waves breaking on a beach 




Fig. 5.1.2. Breaking and diffraction of waves at an inlet 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 71 

in Chapter 10.10 on the basis of other photographs of actual waves 
and a nonlinear treatment of the problem. 

It is clear that the linear theory we apply here can not in principle 
yield large departures from the sine or cosine form of the waves in 
deep water, and still less can it yield the actual breaking phenomena: 
obviously these are nonlinear in character. On the other hand the 
linear theory is to be applied and should yield a good approximation 
for deep water and for the intermediate zone between deep water and 
the actual surf region. However, the fact that breakers do in general 
occur in nature cannot by any means be neglected even in formulating 
the problems in terms of the linear theory, for the following reasons. 
Suppose we consider a train of progressing waves coming from deep 
water in toward shore. As we know from Chapter 3, such a train of 
waves is accompanied by a flow of energy in the direction toward the 
shore. If we assume that there is little or no reflection of the waves 
from the shore— which observations show to be largely the case for a 
gently sloping beach* —it follows that there must exist some mecha- 
nism which absorbs the incoming energy; and that mechanism is of 
course the breaking of the waves which converts the incoming wave 
energy partially into heat through turbulence and partially into the 
energy of a different type of flow, i.e. the undertow. In terms of the 
linear theory about the only expedient which we have at our disposal 
to take account of such an effect in a rough general way is to permit 
that the wave amplitude may become very large at the shore line, or, 
in mathematical terms, that the velocity potential should be per- 
mitted to have an appropriate singularity at the shore line. As we 
have already hinted at the end of the preceding chapter, the ap- 
propriate singularity for a two-dimensional motion seems to be 
logarithmic, and hence the wave amplitude would be logarithmically 
infinite at the shore line. Indeed, it turns out that no progressing 
wave solutions without reflection from the shore line exist at all 
within the framework of the linear theory unless a singularity at 
least as strong as a logarithmic singularity is admitted at the shore 
line. 

The actual procedure works out as follows: Once the frequency 
of the wave motion has been fixed, two different types of standing 

* This fact is also used in laboratory experiments with water waves: the 
experimental tanks are often equipped with a sloping "beach" at one or more 
of the ends in order to absorb the energy of the incoming waves through breaking, 
and thus prevent reflection from the ends of the tank. This makes it possible to 
perform successive experiments without long waits for the motions to subside. 



72 WATER WAVES 

waves are obtained, one of which has finite amplitude, the other 
infinite amplitude, at the shore line. These two different types of 
standing waves behave at oo like the simple standing wave solu- 
tions for water of infinite depth obtained in Chapter 3; i.e. one of 
them behaves like e my sin (mx + a) while the other behaves like 
e my cos ^ mx _|_ a ); hence the two may be combined with appropriate 
time factors to yield arbitrary simple harmonic progressing waves 
at oo. If the amplitude at oo is prescribed, and also the condition 
(cf. the last two sections of the preceding chapter) requiring that the 
wave at oo be a progressing wave moving toward shore, then the 
solution is uniquely determined; in particular, the strength of the 
logarithmic singularity at the shore line is uniquely fixed once the 
amplitude of the incoming wave is prescribed at oo. 

The fact that progressing waves over uniformly sloping beaches 
can be uniquely characterized in the simple way just stated is not 
a thing which has been known for a long time, but represents rather 
an insight gained in relatively recent years (cf. the author's paper 
[S.18] of 1947 and the other references given there). The method 
employed in the author's paper makes essential use of an idea due 
to H. Lewy to obtain the actual solutions for the case of two-dimen- 
sional waves over beaches sloping at the angles nj2n, with n an 
integer; H. Lewy [L.8] extended his method also to solve the problem 
for slope angles (pj2n)7i, with p an odd integer and n any integer 
such that p < 2n. For the special slope angles 7i/2n the progressing 
wave solutions were obtained first by Miche [M.8] (unknown to the 
author at the time because of lack of communications during World 
War II), and somewhat later by Bondi [B.14], but without uniqueness 
statements. Actually, the special standing wave solutions for these 
same slope angles which are finite at the shore line had already 
been obtained by Hanson [H.3]. 

All of these solutions for the slope angles a> = nj2n 9 become more 
complicated and cumbersome as n becomes larger, that is, as the 
beach slope becomes smaller. In fact, the solutions consist of finite 
sums of complex exponentials and exponential integrals, and the 
number of the terms in these sums increases with n. Actual ocean 
beaches usually slope rather gently, so that many of the interesting 
cases are just those in which the slope angle is small— of the order 
of a few degrees, say. It is therefore important to give at least an 
approximate representation of the solution of the problem valid for 
small angles oo independent of the integer n. Such a representation 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 73 

has been given by Friedrichs [F.14]. To derive it the exact solution 
is first obtained for integer n in the form of a single complex integral, 
which can in turn be treated by the saddle point method to yield 
asymptotic solutions valid for large n, that is, for beaches with small 
slopes. The resulting asymptotic representation turns out to be very 
accurate. A comparison with the exact numerical solution for co = 6° 
shows the asymptotic solution to be practically identical with the 
exact solution all the way from infinity to within a distance of less 
than a wave length from the shore line. Eckart [E.2, 3] has devised 
an approximate theory which gives good results in both deep and 
shallow water. 

For slope angles which are rational multiples of a right angle of 
the special form co = p7z/2n with p any odd integer smaller than 2n, 
the problem of progressing waves has been treated by Lewy, as was 
mentioned above. Thus the theory is available for cases in which co 
is greater than n/2, so that the "beach" becomes an overhanging 
cliff. The solution for a special case of this kind, i.e. for co = 135° 
or p = 3, n = 2, has been carried out numerically by E. Isaacson 
[1.2]. It turns out that there is at least one interesting contrast with 
the solutions for waves over beaches in which co < ji/2. In the latter 
case it has been found that as a progressing wave moves in toward 
shore the amplitude first decreases to a value below the value at oo, 
before it increases and becomes very large at the shore line. (This 
fact has also often been verified experimentally in wave tanks). 
The same thing holds for standing waves: at a certain distance from 
shore there exists always a crest which is lower than the crests at go. 
In the case of the overhanging cliff with co = 135°, however, the 
reverse is found to be true: the first maximum going outward from 
the shore line is about 1 % higher than the height of the crests at oo. 
Still another fact regarding the behavior of the solutions near the 
shore line is interesting. In all cases there exists just one standing 
wave solution which has a finite amplitude at the shore line; Lewy 
[L.8] has shown that the ratio of the amplitude there to the am- 
plitude at oo is given in terms of the angle co by the formula (7r/2co) 1/2 . 
Thus for angles co less than tt/2 the amplitude of the standing wave 
with finite amplitude is greater on shore than it is at infinity (becoming 
very large as co becomes small) while for angles co greater than tz/2 
the amplitude on shore is less than it is at oo. Since the observations 
indicate that the standing wave of finite amplitude is likely to be the 
wave which actually occurs in nature for angles co greater than 



74 WATER WAVES " 

about 40°, the above results can be used to give a rational explanation 
for what might be called the "wine glass" effect: wine is much more 
apt to spill over the edge of a glass with an edge which is flared out- 
ward than from a glass with an edge turned over slightly toward the 
inside of the glass. 

A limit case of the problem of the overhanging cliff has a special 
interest, namely the case in which co approaches the value n and the 
problem becomes what might be called the "dock problem": the 
water surface is free up to a certain point but from there on it is 
covered by a rigid horizontal plane. The solutions given by Lewy 
are so complicated as p and n become large that it seems hopeless 
to consider the limit of his solutions as co -> tc. Friedrichs and Lewy 
[F.12] have, however, attacked and solved the dock problem directly 
for two-dimensional waves. For three-dimensional waves in water of 
constant finite depth the problem has been solved by Heins [H.13] 
(also see [H.12]). 

It would be somewhat unsatisfying to have solutions for the sloping 
beach problem only for slope angles which are rational multiples 
of n\ it is clear that this limitation is imposed by the methods used 
to solve the problem and not by any inherent characteristics of the 
problem itself. The two-dimensional problem has, in fact, been solved 
for all slope angles by Isaacson [1.1]. Isaacson obtained an integral 
representation of Lewy's solutions for the angles pn\2n analogous 
to the representation obtained by Friedrichs for the angles nj2n, 
and then observed that his representation depended only upon the 
ratio of p to n and not on these quantities separately. Thus the 
solutions for all angles are given by this representation. Peters [P. 5] 
has solved the same problem by an entirely different method, which 
makes no use of solutions for the special slope angles pn\2n. 

The problem of two-dimensional progressing waves over sloping 
beaches thus has been completely solved as far as the theory of 
waves of small amplitude is concerned. Only one solution for three- 
dimensional motion has been mentioned so far, i.e. the solution by 
Heins for three-dimensional motion in the case of the dock problem. 
For certain slope angles co = nj2n the method used by the author 
[S.18] can be extended in such a way as to solve the problem of 
three-dimensional waves on sloping beaches; in the paper cited the 
solution is carried out for the case co = n\% i.e. for the case of waves 
approaching at an angle and breaking on a vertical cliff. Roseau 
[R.9] has used the same method for the case co = tt/4. Subsequently 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 75 

the problem of three-dimensional waves on sloping beaches has been 
solved by Peters [P. 6] and Roseau [R.9], who make use of a certain 
functional equation derived from a representation of the solution 
by a Laplace integral. In section 5.4. we shall give an account of 
this method of attack. Roseau [R.9] has solved the problem of waves 
in an ocean having different constant depths at the two different 
infinities in the cr-direction which are connected by a bottom of 
variable depth. 

Before outlining the actual contents of the present chapter, it 
may be well to summarize the conclusions which have been obtained 
from studying numerical solutions of the problems being considered 
here, which have been carried out (cf. [S.18]) for two-dimensional 
waves for slope angles co = 135°, 90°, 45°, and 6°, and for three- 
dimensional waves for the case co = 90°. The results for the case of an 
overhanging cliff with co = 135° have already been discussed earlier. 
In the other three cases the most striking and important result is 
the following: The wave lengths and amplitudes change very little 
from their values at oo until points about a wave length from shore 
have been reached. Closer inshore the amplitude becomes large, as 
it must in accord with our theory. It is a curious fact (already men- 
tioned earlier) that the amplitude of a progressing wave becomes 
less (for oj = 6° about 10 % less) at a point near shore than its value 
at oo, although it becomes infinite as the shore is approached. This 
effect has often been observed experimentally. This statement holds 
for the three-dimensional waves against a vertical cliff (with an 
amplitude decrease of about 2 %), as well as for the two-dimensional 
cases. 

The exact numerical solution for the case of a beach sloping at 
6° is useful for the purpose of a comparison with the results obtained 
from the linear shallow water theory (treated in Chapter (10.13) of 
Part III) and from the asymptotic approximation to the exact theory 
obtained by Friedrichs [F.14]. The linear shallow water theory, as 
its name indicates, can in principle not furnish a good approximation 
to the waves on sloping beaches in the deep water portion since it 
yields waves whose amplitude tends to zero at oo. For a beach sloping 
at 6°, for example, it is found that the shallow water theory furnishes 
a good approximation to the exact solution for a distance of two or 
three wave lengths outward from the shore line if the wave length 
is, say, about eight times the maximum depth of the water in this 
range; but the amplitudes furnished by the shallow water theory 



76 WATER WAVES 

would be 50 to 60 percent too small at about 15 wave lengths away 
from the shore line. One of the asymptotic approximations to the 
exact theory given by Friedrichs yields a good approximation over 
practically the whole range from the shore line to infinity (it is in- 
accurate only very close to shore); this approximation, which even 
yields the decrease in amplitude under the value at oo mentioned above, 
is almost identical with one obtained by Rankine (cf. Miche [M.8, 
p. 287]) which is based upon an argument using energy flux con- 
siderations in connection with the assumption that the speed of the 
energy flux can be computed at each point in water of slowly varying 
depth by using the formula (cf. (3.3.9)) which is appropriate in water 
having everywhere the depth at the point in question. Friedrichs thus 
gives a mathematical justification for such a procedure on beaches 
of small slope. 

It has already been made clear that the discussion in this chapter 
cannot yield information about the breaking of waves, which is an 
essentially nonlinear phenomenon. However, it is possible to analyze 
the breaking phenomena in certain cases and within certain limitations 
by making use of the nonlinear shallow water theory, as we shall see 
in Part III. For this purpose, one needs to know in advance the 
motion at some point in shallow water, and this presumably could 
be done by using the methods of the present chapter, combined 
possibly with the methods provided by the linear shallow water 
theory. 

The material in the subsequent sections of this chapter is ordered 
as follows. In section 5.2. the problem of two-dimensional progressing 
waves over beaches sloping at the angles 7t/2n, n an integer, is discussed 
following the method of Lewy [L.8] and the author [S.18]. In section 
5.3 the problem of three-dimensional waves against a vertical cliff 
is treated, also using the author's method. The reasons for including 
these treatments in spite of the fact that they yield results that are 
included in the more general treatments of Peters [P. 6] and Roseau 
[R.9] is that they are interesting in themselves as an example of 
method, and also they can be applied to other problems, such as the 
problem of plane barriers inclined at the angles 7i/2n (cf. F. John 
[J. 4]), which have not been treated by other methods. In section 5.4, 
the general problem of three-dimensional waves on beaches sloping 
at any angle is treated following essentially the ideas of Peters. 

In section 5.5 the problem of diffraction of waves around a rigid 
vertical wedge is treated; in case the wedge reduces to a plane the 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



77 






problem becomes the classical diffraction problem of Sommerfeld 
[S.12] for the case of diffraction of plane waves in two dimensions 
around a half-plane barrier. A new uniqueness theorem and a new and 
elementary solution for the problem are given. Methods of analyzing 
the solution are also discussed; photographs of the waves in such cases 
and comparisons of theory and experiment are made. 

Finally, in section 5.6 a brief survey of a variety of solved and 
unsolved problems which might have been included in this chapter, 
with references to the literature, is given. Included are brief references 
to researches in oceanography, seismology, and to a selection of papers 
dealing with simple harmonic waves by using mathematical methods 
different from those employed otherwise in this chapter. In parti- 
cular, a number of papers employing integral equations as a basic 
mathematical tool are mentioned and occasion is taken to explain 
the Wiener-Hopf technique of solving certain singular integral 
equations. 



5.2. Two-dimensional waves over beaches sloping at angles a)=ji/2n 

We consider first the problem of two-dimensional progressing 
waves over a beach sloping at the angle co = 7i/2n with n an integer 




Fig. 5.2.1. Sloping beach problem 

(cf. Figure 5.2.1), in spite of the fact that the problem can be solved, 
as was mentioned in the preceding section, by a method which is not 



78 WATER WAVES 

restricted to special angles (cf. Peters [P. 6], and Roseau [R.9]). 
The problem is solved here by a method which makes essential use of 
the fact that the slope angle has the special values indicated because 
the method has some interest in itself, and it yields representations 
which have been evaluated numerically in certain cases. In addition, 
the relevant uniqueness theorems are obtained in a very natural way. 
We assume that the velocity potential &(x, y; t) is taken in the 
form = e iot (p(x, y). Hence cp(x, y) is a harmonic function in the 
sector of angle co = 7i/2n. The free surface boundary condition then 
takes the form 

G 2 

(5.2.1) <Pv——<p = 9 fort/ = 0, x > 0, 

% 

as we have often seen (cf. (3.1.7)), while the condition at the bottom is 

(5.2.2) ^ = 0. 

dn 

It is useful to introduce the same dimensionless independent variables 
as were used in the preceding chapter: 

(5.2.3) x x = mx, y Y = my, m = o 2 /g. 

The function cp(x, y) obviously remains harmonic in these variables, 
and conditions (5.2.1) and (5.2.2) become 

(5.2.1)' cp y -(p = 0, y = 0, x>0, 

(5.2.2)' cp n = 0, 

after dropping subscripts. 

The simple harmonic standing waves in water of infinite depth 
everywhere are given by 

cos (x + a) 



(5.2.4) 0(x, y; t) = e iat • e* 

1 sin (x ~\- ol, 

we write these down because we expect that they will represent the 
behavior of the standing waves in our case at large distances from 
the origin, that is, far away from the shore line. 

The solution of the problem is obtained in terms of the complex 
potential f(z) defined by 

(5.2.5) f(z) = f(x + iy) = <p(cc, y) + i%(x, y). 

The function f(z) should, like (p, be regular and analytic in the 



:«) 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 79 

entire sector (including the boundaries,* except for the origin). 
The boundary conditions (5.2.1)' and (5.2.2)' are given in terms of 

/(«) by 

(5.2.6) tp m - <p = me Q- - i\ (y + %x) = me (- - l) /(> 

= me (if — /) = for z real and positive, 
pi 

(5.2.7) w n = me— (/(»)) = me (- iexp (- inj2n) /') 

= for 2 = r exp {— ijifen}, r > 0. 
The second condition results from 

5n 1 r dd ) 

We introduce the two following linear differential operators: 

(5.2.8) L^D) = - i exp {- in/2n} D, 

(5.2.9) L 2n (D) = iD — 1 

with D meaning rf/c?z. The basic idea of the method invented by 
H. Lewy is to find additional linear operators, L 2 , L 3 , . . ., L 2n _ 1 
such that the operation L x • L 2 • . . . • L 2n applied on f(z) yields a 
function F(z) whose real part vanishes on both boundaries of our sector. 
Once this has been done, the function F(z) can be continued analyti- 
cally over the boundaries of the sector by successive reflections to 
yield a single-valued function defined in the entire complex plane 
except possibly the origin. It can then be shown (see [S.18]), essen- 
tially by using Liouville's theorem, that the function F(z) is uniquely 
determined within a constant multiplying factor by boundedness 
conditions on the complex potential f(z) at oo together with the order 
of the singularity admitted at the origin. After F(z) has been thus 
determined, the complex potential f(z) is obtained as a solution of 
the ordinary differential equation L X L 2 . . . L 2n f(z) = F(z). Of course, 
it is necessary in the end to determine the arbitrary constants in the 
general solution of this differential equation in such a way as to 
satisfy all conditions of the problem, and this can in fact be done 
explicitly. It turns out that the resulting solution behaves at oo like 

* Far less stringent conditions at the boundaries could be prescribed, since 
analytic continuations over the boundaries can easily be obtained explicitly in 
the present case. 



80 WATER WAVES , 

the known solutions for waves in water having infinite depth every- 
where and that it is uniquely determined by prescribing the amplitude 
of the wave at oo together with the assumption that it should be, 
say, an incoming wave. 

We proceed to carry out this program, without however giving 
all of the details (which can be found in the author's paper [S.18]). 
To begin with, the ordinary differential equation for f(z) and the 
operators L t are given by 

(5.2.10) L(D)f = L 1 L 2 L 3 -...- L,J 

= (a 1 D)(a 2 fl - l)(a 3 Z))(a 4 D - 1) . . . (a 2 „_ 1 Z))(a 2B Z> - 1)/ 
= F(z) 

with the complex constants a^ defined by 

f k 1\ 

(5.2.11 ) a. k = e~ in \*i + 2) , k = 1, 2, . . ., 2n. 

One observes that L X (D) and L 2n (D) coincide with the definitions 
given in (5.2.8) and (5.2.9). It is, in fact, not difficult to verify that 

(5.2.12) &eF(z) = 

on both boundaries of the sector, by making use of the properties 
of the numbers 0L k and of the fact that 0te L ± (D) and 0te L 2n (D)f 
vanish on the bottom and the free surface, respectively, by virtue 
of the boundary conditions (5.2.7) and (5.2.6). 

So far we have not prescribed conditions on f(z) at oo and at the 
origin, and we now proceed to do so. At the origin we assume, in 
accordance with the remarks made in section 5.1 and the discussion 
in the last section of the preceding chapter, that f(z) has at most a 
logarithmic singularity; we interpret this to mean that | d k f{z)jdz k \ < 
M k /\ z \ k in a neighborhood of the origin for k = 1, 2, . . ., 2n, with 
M k certain constants. At oo we require that cp = &e f(z) together 
with | d k f(z)jdz k | for k = 1, 2, . . ., 2n be uniformly bounded when 
z -> oo in the sector. (These conditions could be weakened con- 
siderably, but they are convenient and are satisfied by the solutions 
we obtain. ) In other words, although we expect the solutions of our 
problem to behave at oo in accordance with (5.2.4) it is not necessary 
to prescribe the behavior at oo so precisely since the boundedness 
conditions yield solutions having this property automatically. Once 
these conditions on f(z) have been prescribed we see that the function 
F(z) defined by (5.2.10) has the following properties: 1) | F(z) | is 






WAVES ON SLOPING BEACHES AND PAST OBSTACLES 81 

uniformly bounded in the sector, and 2) \ F(z) \ = 0(l/z 2n ) in the 
neighborhood of the origin. 

We have already observed that 3$e F(z) = on both boundaries 
of the sector and that F(z) can therefore be continued as a single- 
valued function into the whole plane, except the origin, by the 
reflection process. Here we make decisive use of the assumption 
that co, the angle of the sector, is 7i/2n with n an integer. Since the 
boundedness properties of F(z) at oo and the origin are preserved 
in the reflection process, it is clear from well-known results concerning 
analytic functions that F(z) is an analytic function over the whole 
plane having a pole of order at most 2n at the origin. Since in ad- 
dition the real part of F(z) vanishes on all rays z = r exp{i&7r/2n}, 
k = 1, 2, . . ., 4n, it follows that F(z) is given uniquely by 

(5.2.13) *■(*) = %* 

z 2n 

with A 2n an arbitrary real constant which may in particular have 
the value zero. Thus the complex potential f(z) we seek satisfies the 
differential equation 

(5.2.14) ( ai Z))(a 2 Z) - 1 ) . . . (a^flJM - 1 )/ = %*. 

z 2n 

Our problem is reduced to finding a solution f(z) of this differential 
equation which satisfies all of the conditions imposed on f(z). From 
the discussion of section 5.1 we expect to find two solutions f^z) 
and f 2 (z) of our problem which behave differently at the origin and 
at oo; at the origin, in particular, we expect to find one solution, 
say fx(z), to be bounded and the other, f 2 (z), to have a logarithmic 
singularity. 

The regular solution f x (z) is the solution of (5.2.14) which one 
obtains by taking for the real constant A 2n the value zero, while 
f 2 (z) results for A 2n ^ 0. In other words the solution of the non- 
homogeneous equation contains the desired singularity at the origin. 
One finds for f x (z) the solution 

(5.2.15) f ± (z) = U V c k e z \ 

(n-l)!Vn£i 

in which the constants c k and {3 k are the following complex numbers: 



82 



WATER WAVES 



(5.2.16 



P k = exp in 



Cv. 



exp 



U+i) 



n 2n 
cot — cot 



. cot 



(Jc-I)ti 



2n 



2n 



2n 
k = 2,3, ...,n 



The constants c k are obtained by adjusting the arbitrary constants 
in the solution of (5.2.14) so that the boundary conditions on f(z) 
at the free surface and the bottom are satisfied; that such a result 
can be achieved by choosing a finite number of constants is at first 
sight rather startling, but it must be possible if it is true that a func- 
tion f(z) having the postulated properties exists since such a function 
must satisfy the differential equation (5.2.14). The calculation of the 
constants c k is straightforward, but not entirely trivial. The function 
f y (z) is uniquely given by (5.2.15) within a real multiplying factor. 
As | z | -> oo in the sector, all terms clearly die out exponentially 
except the term for k = n, which is c n exp {— iz}, since all /? fc 's 
except fi n have negative real parts. Even the term for k = n dies 
out exponentially except along lines parallel to the real axis. (The 
value of c n , by the way, is exp {— ijt(n — l)/4} since the cotangents 
in (5.2.16) cancel each other for k = n.) This term thus yields the 
asymptotic behavior of f x (z): 

(5.2.17) f x (z) ~ ^- 7 --- • c n er*>. 

The solution f 2 (z) of the nonhomogeneous equation (5.2.14) which 
satisfies the boundary conditions is as follows: 



(5.2.18 



/»(*) = 2 a* 



k=l 



i 



izP k e it 



dt 



me 



z p k 



for the case in which the real constant A 2n is set equal to one. The 
constants /3 k are defined in (5.2.16); and the constants a k are defined by 

(5.2.19) a k = c k /{(n - l)Wn}, 

that is, they are a fixed multiple (for given n) of the constants c k 
defined in (5.2.16). The constants a k , like the c k , are uniquely deter- 
mined within a real multiplying factor. The path of integration for 
all integrals in (5.2.18) is indicated in Figure 5.2.2. That the points 
izf$ k lie in the lower half of the complex plane (as indicated in the 
figure) can be seen from our definition of the constants j3 k and the 
fact that z is restricted to the sector — n\2n ^ arg z ^ 0. 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



83 




Fig. 5.2.2. Path of integration in /-plane 



The behavior of f 2 (z) at oo of course depends on the behavior 
of the functions in (5.2.18). It is not hard to show— for example, 
by the procedure used in arriving at the result given by (4.3.15) 
in the preceding chapter — that these functions behave asymptotically 
as follows: 



riePje e it 

(5.2.20) e g Pk\ —dt 

J zoo * 



o(\) , 0te (izfi k ) < 0, Jm (izp k ) ^ 0, 

2ni-o(\) , 0te (izp k ) > 0, Jm (izfi k ) ^ 0. 

Once this fact is established it is clear from (5.2.19) and (5.2.18) 
that / 2 (jz) behaves asymptotically as follows: 



(5.2.21) 



/.(*) 



n 



[n — 1)1 y/n 



c„ie~ 



since the term for k = n dominates all others (cf. (5.2.20)) and 
&e{iz[$ k ) > in this case. Comparison of (5.2.21) with (5.2.17) shows 
that the real parts of f^z) and f 2 (z) would be 90° out of phase at oo. 
That the derivatives of / 2 (2) behave asymptotically in the same 
fashion as f 2 (z) itself is easily seen, since the only terms in the deriva- 
tives of (5.2.18) of a type different from those in (5.2.18) itself are 
of the form b k /z k , k an integer ^ 1. Finally, it is clear that f 2 (z) 
has a logarithmic singularity at the origin. Hence f^z) and f 2 (z) 
satisfy all requirements. Just as in the 90° case (cf. the last section 



84 WATER WAVES 

of the preceding chapter) it is now clear that f(z) = b-J x (z) + b 2 f 2 (z), 
with b x and b 2 any real constants, yields all standing wave solutions 
of our problem. 

The relations (5.2.17) and (5.2.21 ) yield for the asymptotic behavior 
of the real potential functions cp 1 and cp 2 the relations: 

jl I yi 1 \ 

(5.2.22) fp^y) = &eli~- — — - e'cos [x + — — n\ 

(5.2.23) cp 2 (x, y) = <%ef 2 ~ ^— — e* sin (x + !Lzi „\ 

(n — l)\\/n \ 4 / 

when it is observed that c n = exp {— in{n — l)/4}. It is now possible 
to construct either standing wave or progressing wave solutions which 
behave at oo like the known solutions for steady progressing waves 
in water which is everywhere infinite in depth. In particular we 
observe that it makes sense to speak of the wave length at oo in our 
cases and that the relation between wave length and frequency 
satisfies asymptotically the relation which holds everywhere in water 
of infinite depth. For this, it is only necessary to reintroduce the 
original space variables by replacing x and y by mx and my, with 
m = o 2 /g (cf. (5.2.3)), and to take note of (5.2.22) and (5.2.23). 

Finally, we write down a solution @(x, y; t) which behaves at oo 
like e v cos (x + t + a), i.e. like a steady progressing wave moving 
toward shore: 

(5.2.24) 0(x, y; t) = Afy^x, y) cos (t + a) - <p 2 (x, y) sin (t + a)]. 

As our discussion shows, this solution is uniquely determined as soon 
as the amplitude is prescribed at go (i.e. as soon as A is fixed) since 
q>i(x, y) and cp 2 (x, y) yield the only standing wave solutions of our 
problem and they are determined also within a real factor. As we 
have already stated in the preceding section, the progressing wave 
solutions (5.2.24) have been determined numerically (cf. [S.18]) for 
slope angles co = 90°, 45°, and 6°, with results whose general features 
were already discussed in that section. 

5.3. Three-dimensional waves against a vertical cliff 

It is possible to treat some three-dimensional problems of waves 
over sloping beaches by a method similar to the method used in the 
preceding section for two-dimensional waves, in spite of the fact 
that it is now no longer possible to make use of the theory of analytic 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 85 

functions of a complex variable. In this section we illustrate the 
method by treating the problem of progressing waves in an infinite 
ocean bounded on one side by a vertical cliff when the wave crests 
at oo may make any angle with the shore line (cf. [S.18]). 

We seek solutions 0(x, y, z; t) of V 2 (iC y z) = in the region 
x ^ 0, y ^ 0, — oo < 2 < co with the y-axis taken normal to the. 
undisturbed free surface of the water and the z-axis* taken along 
the "shore", i.e. at the water line on the vertical cliff x = 0. Progressing 
waves moving toward shore are to be found such that the wave 
crests (or other curves of constant phase) at large distances from 
shore tend to a straight line which makes an arbitrary angle with 
the shore line. For this purpose we seek solutions of the form 

(5.3.1) 0(x, y, z; t) = exp {i(at + kz + ^)}(p(x, y) 

that is, solutions in which periodic factors in both z and t are split off. 
As in the preceding section, we introduce new variables and para- 
meters through the relations x x = mx, y x = my, % = mz, k 1 — k/m, 
m = G 2 /g and obtain for cp the differential equation 

(5.3.2) Vf*,y)<P — k *<P =~ ° 
and the free surface condition 

(5.3.3) <p y —fp = Q for y = 0, 

after dropping the subscript 1 on all quantities. The condition at 
the cliff is, of course, 

(5.3.4) -^ = for x = 0. 

ox 

At the origin x — 0, y — (i.e. at the shore line on the cliff) we 
require, as in former cases, that cp should be of the form 

(5.3.5) (p =■■ <p log r -\-<p, r <1, 

for sufficiently small values of r = (x 2 -j- y 2 ) 112 . with ^ and q> certain 
bounded functions with bounded first and second derivatives in a 
neighborhood of the origin. The functions ~cp and 7p should be considered 
at present as certain given functions; later on, they will be chosen 
specifically. 

For large values of r we wish to have 0(x, y, z; t) behave like 

* It has already been pointed out that functions of a complex variable are 
not used in this section, so that the reintroduction of the letter z to represent a 
space coordinate should cause no confusion with the use of the letter 2 as a complex 
variable in earlier sections. 



86 WATER WAVES 

e y exp {i(ot + kz + olx + /?)} with A; 2 + a 2 = 1 but & and a otherwise 
arbitrary constants, so that progressing waves tending to an arbitrary 
plane wave at oo can be obtained. This requires that (p(x, y) should 
behave at oo like e v exp {i(<x.x + /? 2 )} because of (5.3.1). However, 
it is no more necessary here than it was in our former cases to require 
that cp should behave in this specific way at oo; it suffices in fact 
to require that 



(5.3.6) | <p | + I (Px I + I <Pxy I < M for r> R t 



0' 



i.e. that <p and the two derivatives of cp occurring in (5.3.6) should 
be uniformly bounded at oo. As we shall see, this requirement leads 
to solutions of the desired type. 

We proceed to solve the boundary value problem formulated in 
equations (5.3.2) to (5.3.6). The procedure we follow is analogous 
to that used in the two-dimensional cases in every respect. To begin 
with, we observe that 



dx \dy ) 



(5.3.7) — U- — 1 )<P = ° for both x = and y == 0, 

ox \dy J 

because of the special form of the linear operator on the left hand 
side together with the fact that (5.3.3) and (5.3.4) are to be satisfied. 
A function ip(x, y) is introduced by the relation 



- \~- 



d_/d_ 

dx \dy 



The essential point of our method is that the function ip is determined 
uniquely within an arbitrary factor if our function (p, having the 
properties postulated, exists. Furthermore, ip can then be given explic- 
itly without difficulty. The properties of ip are as follows. 

1. ip satisfies the same differential equation as 99, i.e. equation 
(5.3.2), as one sees from the definition (5.3.8) of ip. 

2. ip is regular in the quadrant x > 0, y < and vanishes, in view 
of (5.3.7), on x = 0, y < and y = 0, x > 0. Hence ip can be 
continued over the boundaries by the reflection process to yield a 
continuous and single-valued function having continuous second 
derivatives ip xx and ip yy (as one can readily see since V 2 ^ — k 2 ip = 0, 
and ip = on the boundaries) in the entire x, y-plane with the ex- 
ception of the origin. (Here we use the fact that our domain is a sector 
of angle tt/2.) 






WAVES ON SLOPING BEACHES AND PAST OBSTACLES 87 

3. At the origin, ip has a possible singularity which is of the form 
<p(x, y)/r 2 , with ~cp regular, as one can see from (5.3.5) and (5.3.8). 
This statement clearly holds for the function ip when it has been 
extended by reflection to a full neighborhood of the origin. 

4. The condition (5.3.6) on cp clearly yields for ip the condition 
that ip is uniformly bounded at oo after ip has been extended to the 
whole plane. 

Thus ip is a solution of V 2 ip — k 2 ip — in the entire plane which 
is uniformly bounded at oo. At the origin ip = y/r 2 + ~cp with <p and 
q> certain regular functions (^ = not excluded). In addition, ip = 
on the entire x and y axes. We shall show, following Weinstein 
[W.5],* that the function 

(5.3.9) ip(x, y) = AM™ (ikr) sin 26, r == Vx 2 + y 2 , ^ k ^ 1 

is the unique solution for y in polar coordinates (r, Q) with A an 
arbitrary real constant, and H^ the Hankel function of order two 
which tends to zero as r -> oo. The function ip has real values for r 
real. (The notation given in Jahnke-Emde, Tables of Functions, is 
used.) 

The solution ip is obtained by Weinstein in the following way. 
In polar coordinates (r, 6) the differential equation for ip is 

d 2 ip 1 dw I d 2 ip 

■z-?- + - "ZT + — k 2 ip = 0. 

dr 2 r dr r 2 dd 2 

For any fixed value of r the function ip can be developed in the 
following sine series: 

00 

ip = 2 C n( r ) Sm 2^0 

since ip vanishes for 6 = 0, tz/2, n, Sjz/2; and the coefficients c n (r) 
are given by 

c n (r) = C n r' 2 ip(r, 6) sin 2n6 d6, n = 1, 2, . . ., 

Jo 

with C n a normalizing factor. From this formula one finds by differen- 
tiations with respect to r and use of the differential equation for ip 
that c n (r) satisfies the equation 



"(r) 

n v ' 



1 / 4t? 2 \ C r 71 ! 2 /?) 2 ™ \ 

- <M) - (V + -j) Cn(r) = - -J ^ (g + 4»V) sin 2«6 dO. 



* In the author's paper the solution ip was obtained, but with a less general 
uniqueness statement. 



88 WATER WAVES 

The right hand side of this equation vanishes, as can be seen by 
integrating the first term twice by parts and making use of the 
boundary conditions yj = for 6 = and 6 = n/2. Thus the functions 
c n (r) are Bessel functions, as follows: 

c n (r) = A 2n i^H ( V(ikr) + B 2n I 2n (kr), 

with A 2n and B 2n arbitrary real constants. The functions I 2n are 
unbounded at oo; the Hankel functions H^ behave like r~ 2n for 
r -> and tend to zero exponentially at oo. It follows therefore that 
the Fourier series for ip in our case reduces to the single term given 
by (5.3.9) because of the boundedness assumptions on ip. 

For our purposes it is of advantage to write the solution xp in the 
following form: 

a 2 



(5.3.10) w = Ai — - H® (ikr), r = Vx 2 + y 2 , 

decoy 

in which A is any real constant and H® is the Hankel function of 
order zero which is bounded as r -> oo. It is readily verified that this 
solution differs from that given by (5.3.9) only by a constant multi- 
plier: for example, by using the well-known identities involving the 
derivatives of Bessel functions of different orders. 

Once ip is determined we may write (5.3.8) in the form 

(5.3.11) — — - 1 )<p = Ai—- H® (ikr), A arbitrary. 

dec \dy ) dxdy 

This means that our function 99, if it exists, must satisfy (5.3.11) as 
well as (5.3.2). By integration of (5.3.11) it turns out that we are 
able to determine 99 explicitly without great difficulty on account 
of the simple form of the left hand side of (5.3.11). This we proceed 
to do. 

Integration of both sides of (5.3.11) with respect to x leads to 

(5.3.12) (A--l\<p = Ai?- H® (ikr) + g(y), 

in which g(y) is an arbitrary function. But g(y) must satisfy (5.3.2), 
since all other terms in (5.3.12) satisfy it. Hence d 2 g/dy 2 — k 2 g = 0. 
In addition g(0) = 0, since the other terms in (5.3.12) vanish for 
y = because of (5. 3. 3) and the fact that dH® ldy={ik)- 1 (yjr)dH^ ) /dr. 
Finally, g(y) is bounded as y -> — 00 because of condition (5.3.6) 
and the fact that dH^ jdy tends to zero as r -> 00. The function 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 89 

g(y) is therefore readily seen to be identically zero. By integration 
of (5.3.12) we obtain (after setting g(y) = 0): 

(5.3.13) cp = Aiey e~* — [H { V (ikVx 2 + t 2 )]dt + B(xje*. 

J + oo 01 

The function B(x) and the real constant A are arbitrary. The integral 
converges, since d{H { ^ )/dt dies out exponentially as t -> oo. 

We shall see that two solutions (p^x, y) and cp 2 (x, y) satisfying all 
conditions of our problem can be obtained from (5.3.13) by taking 
A = in one case and A ^ in the other case, and that these 
solutions will be 90° "out of phase" at oo. (This is exactly analogous 
to the behavior of the solutions in our previous two-dimensional cases. ) 
Consider first the case A = 0. The function cp given by (5.3.13) 
satisfies (5.3.2) only if 

(5.3.14) ^^ + (1 - k 2 )B(x) = 0. 

V ' dx 2 V 

It is important to recall that k 2 < 1. The boundary condition 
(p x = for x = requires that B x (0) = 0. The condition cp y — (p = 
for y = is automatically satisfied because of (5.3.12) and g(y) = 0. 
Hence B(x) = A x cos vl — k 2 x, with^ arbitrary, and the solution 
(p ± (x, y) is 



(5.3.15) (p x (x, y) = A t ev cos Vl — k 2 x. 

This leads to solutions ± in the form of standing waves,* as follows: 

sin kz 

for k 2 < 1. If k = 1, the solution 1 given by (5.3.15)' continues to 
be valid. 

As we have already stated, we obtain solutions (p 2 (x, y) from 
(5.3.13) for A ^ which behave for large x like sin Vl — k 2 x rather 
than like cos Vl — k 2 x, and with these two types of solutions 
progressing waves approaching an arbitrary plane wave at oo can 
be constructed by superposition. 

We begin by showing that (5.3.2) is satisfied for all x > 0, y < 
by cp as given in (5.3.13) with A ^ 0, provided only that B(x) 

* The standing wave solutions of this type (but not of the type with a singu- 
larity) for beaches sloping at angles 7i/2n were obtained by Hanson [H.3] by a 
quite different method. 



90 



WATER WAVES 



satisfies (5.3.14). Since x > 0, it is permissible to differentiate under 
the integral sign in (5.3.13), even though t takes on the value zero 
(since the upper limit y is negative). By differentiating we obtain 



(5.3.16) V 2 cp- k 2 <p = Aile 



1 

J 00 



dt 



dx* 



+ (1 



ff»A 



+ 



dy 



d 2 H^ 



dy' 



+ {B"(x) + (1 -k 2 )B(x)}ey. 



Since HM 



is a solution of (5.3.2) the operator (d 2 /dx 2 — k 2 ) oc- 
curring under the integral sign can be replaced by — d 2 /dy 2 and hence 
the integral can be written in the form 



J oo L dt^dt_ 



H™ (ikr)dt. 



We introduce the following notation 



i m (x> y) 



e,y 



/• v 

I ' 

J 00 



ar 



ffM (*fcr)# 



and obtain through two integrations by parts the result 



^m(«» 2/) 



a^- 1 a?/ m - 2 _ 



JT(D 
^0 



+ ^ 






a, 



in which we have made use of the fact that the boundary terms are 
zero at the lower limit + oo, since all derivatives of H^ (ikr) tend 
to zero as r -> + oo. The integral of interest to us is given obviously 
by I 1 — I z and this in turn is given by 



^3+^! 



d 2 H^ 



OT w 



dy 



+ «i 



^ 



I ■ 

»/ 00 



dt 



dt 



J ' 

»/ 00 



a/ 



d 2 H^ 



mp 



dy 2 dy 

by use of the above relations for I m . Hence the quantity in the first 
bracket in (5.3.16) is identically zero— in other words the term 
containing the integral on the right hand side of (5.3.13) is a solution 
of (5.3.2). Hence cp is a solution of (5.3.2) in the case A ^ if 
B(x) satisfies (5.3.14). Since (5.3.12) holds and g(y) = it follows 
that the free surface condition (5.3.3) is satisfied by <p in view of 
the fact that dH^ (ikr)jdy = for y = 0. 

We have still to show that a solution B(x) of (5.3.14) can be chosen 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



91 



so that cp x = for x = 0, and that cp has the desired behavior for 
large values of r. Actually, these two things go hand in hand. An 
integration by parts in (5.3.13) yields the following for <p: 



(5. 



5.3.17) f=Aie* f e~'H^ (ikVx 2 +fi)dt+AiH^ (ikVx 2 +y 2 )+B(x)ey, 

J 00 

provided that x > 0. It should be recalled that the upper limit y 
of the integral is negative; thus the integrand has a singularity for 
x = since t = is included in the interval of integration and 
Hl 1J (ikr) is singular for r = 0. We shall show that lim dcp/dx = 



provided that B x (0) = - 2 A ^ 0. We have, for x > and y < 0: 

a; 



? = Aie" f V 1 — [ffW (ifcVa; 2 + * 2 ")]d* 

'a J oo da? 



. a 



+ ^ — [HW (ihVx* + i/ 2 )] + B x (x)ey. 
ox 

The second term on the right hand side is readily seen to approach 
zero as x -> since this term can be written as the product of x and 
a factor which is bounded for y < 0. For the same reason it is clear 
that the only contribution furnished by the integral in the limit 
as x -> arises from a neighborhood of t = since the factor x may 
be taken outside of the integral sign. We therefore consider the limit 

e , a 



nm 

05->0 



dx 



[iffW (ifcV^ 2 + * 2 )]d£, e > 0. 



The function iH^ ] (ikr) has the following development valid near 
r = 0: 

2 
iff^ (ikr) — — - [J (ikr) log r + p(r)] 

in which p(r) represents a convergent power series containing only 
even powers of r, and J is the regular Bessel function with the 
following development 



J (ikr) 


(At) 2 

- 1 + + . . .. 
2 2 




It follows that 


1 [iffg) (Or)] = - - 

<7£ 7T 


- J (ikr) + Ji(iAr) - log r + /rg(r) 
_/"* r 


2 


~ x 1 

— J (ikr) + - /c 2 « log r + xg(r) 
r 2 2 





92 WATER WAVES , 

in which g(r) = (l/r)dp/dr is bounded as x^O since y<0. The con- 
tribution of our integral in the limit is therefore easily seen to be 
given by 



2 f- £ , x 2 r- e x 

lim — - e-t— dt = lim — - — — 

x ^ 71 J e X 2 + t 2 x ^ 71 J £ X 2 + 



- dt. 
t 2 



By introducing u = t/x as new integration variable and passing to 
the limit we may write 

2 f- £ x 2 T- 00 dw 
lim - - dt = - - = 2. 

x ^ 7lJ E X 2 + t 2 TlJ^l+U 2 

It therefore follows that lim dcp/dx = provided that 

(5.3.18) B x (0) = — 2A. 

The function B{x) which satisfies this condition and the differential 
equation (5.3.14) is 

(5.3.19) B(x) = — — == sin Vl — k 2 x. 

Vl - k 2 

Since H^ (ikr) dies out exponentially as r -> oo it follows that the 
solution 99 given by (5.3.17) with B(x) defined by (5.3.19) behaves 
at 00 like e v sin [(1 — k 2 ) ll2 x]. 

A solution 9? 2 of our problem which is out of phase with 9^ (cf. 
(5.3.15)) is therefore given by 



(5.3.20) <p 2 (x, y) = A 



it"- 



e-m^ (ikVx 2 + t 2 )dt 



2e y 



+ iH^ (ikVx 2 + y 2 ) - —= sin Vl - k 2 x 

Vl-k 2 

with A 2 an arbitrary real constant. Standing wave solutions & 2 are 
then given by 

cos kz 



(5.3.20)' 2 = A 2 e iot <p 2 (x, y) 

sin kz 

By taking appropriate values of k progressing waves tending at 00 
to any arbitrary plane wave solution for water of infinite depth can 
be obtained by forming proper linear combinations of solutions of the 
type (5.3.15)' and (5.3.20)'. For a progressing wave traveling toward 
shore, for example, we may write 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



93 



(5.3.21) 0{x, y, z;t)=A 



(p ± {x, y)cos kz 



cp^x, ?/)sin kz 



Vi-k 2 



Vi-¥ 



(p 2 {x, y) sin kz 



cp 2 (x, y ) cos kz 



cos at 



sin ot 



in which A 1 and A 2 in (5.3.15) and (5.3.20) are both taken equal 
toX The solution (5.3.21 ) behaves at oo \ikeAe y cos ( Vl —k 2 x +kz+ot) 
as one can readily verify by making use of the asymptotic behavior 
of <p x (x, y) and cp 2 (x, y)* and it is the only such solution since <p 3 
and (p 2 are uniquely determined. 

The special case k = 1 has a certain interest. It corresponds to 
waves which at oo have their crests at right angles to the shore. 
One readily sees from (5.3.15) and (5.3.20) that as k -> 1 the pro- 
gressing wave solution (5.3.21) tends to 

(5.3.22) 0(y, z; t) = Ae v cos (z + at) 

that is, the progressing wave solution for this case is independent 
of x, is free of a singularity at the origin, and the curves of constant 
phase are straight lines at right angles to the shore line — all properties 
that are to be expected. 

The progressing wave solution (5.3.21) was studied numerically 



(x,0) 
































/ 




































/ 




n 


* 


D 


























/ 











\ 


























/ 








\ 


\ 
























/ 





































/ 








-i 




\ 


\ 




















/ 
















\ 


\ 
















/ 


/ 
















\ 


s\ 
















/ 




















V 


>r 


* 2 (x,o) 






/ 














-2 










^ 










J 




























^ 


^ 




s? 


*2_. In (/£?.)_< 






























































?> 





































Fig. 5.3.1. Standing wave solution for a vertical cliff (with crests at an angle 

of 30° to shore) 



* We remark once more that the original space and time variables can be 
reintroduced simply by replacing x, y, z by mx, my, mz and k by k/m. 



94 



WATER WAVES 



for k = 1/2, i.e. for the case in which the wave crests tend at oc 
to a straight line inclined at 30° to the shore line. The function 
<p 2 (x, 0) is plotted in Figure 5.3.1. With the aid of these values the 
contours for were calculated and are given in Figure 5.3.2. These 
are also essentially contour lines for the free surface elevation r\, 

in accordance with the formula r\ = t | ^=0 . The water surface 

is shown between a pair of successive "nodes" of 0, that is, curves 
for which = 0. These curves go into the z-axis (the shore line) 
under zero angle, as do all other contour lines. This is seen at once 
from their equation (cf. (5.3.21) with at = ji/2) 



(5.3.23) (p x (x y 0) cos kz + 



VI -k 2 



(p 2 (x, 0) sin kz = const. 



Since <p 2 -> oo as x -> while cp x remains bounded, it is clear that 
sin kz must approach zero as x -> on any such curve. That the 
contours are all tangent to the 2-axis at the points z = 2jin, n an 
integer, is also readily seen. It is interesting to observe that the 



12 


417 




































V 




































in 


N. 






































w 


\ 


































8 


\v 




































\ 








vJ 




























6 


A 


•\ S 










1 






















Sq 


E> = -0.9 


8^ 








^T"^ 




















4 












p>v 
























\ 


i\ 








£ = 


nc 


^ 




















2 






N 


























-4 


=0.( 


) 


















, j^ 
















>=-0.5 

' = •0.9 

5>«-i'o 

1 
= -0.9 

1 
>=-0.5 


o 
















^^ 












































^ 


5*3 


-2 
































^ 


































^<J 


-4 
































--<J 


>:( 


).0 

- 1 >■ 



I 2 3 4 x 

Fig. 5.3.2. Level lines for a wave approaching a vertical cliff at an angle 

height of the wave crest is lower at some points near to the cliff than 
it is at oo. It may be that the wave crest is a ridge with a number 
of saddle points. 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



95 



It should be pointed out that we are no more able to decide in the 
present case than we were in the two-dimensional cases whether 
the waves are reflected back to infinity from the shore, and if so to 
what extent. Our numerical solution was obtained on the assumption 
that no reflection takes place, which is probably not well justified 
for the case of a vertical cliff, but would be for a beach of small slope. 



5.4. Waves on sloping beaches. General case 

We discuss here the most general case of periodic waves on sloping 
beaches which behave at oo like an arbitrary progressing wave — in 
particular, a wave with crests at an arbitrary angle to the shore line— 
and for a beach sloping at any angle. As has been mentioned earlier, 
this problem was first solved by Peters and Roseau (cf. the remarks 
in section 5.1). 

We seek a harmonic function 0(x, y,z; t) of the form exp {i(ot-\-kz)} 
• (p(x, y) in the region indicated in cross section in Figure 5.4.1. At 




Fig. 5.4.1. Sloping beach of arbitrary angle 



oo the function should behave like exp {i(ot-\-kz-\-0Lx)} ■ exp {o 2 y/g} 
with k and a arbitrary. The function cp (x, y) is not a harmonic func- 
tion, but satisfies, as one readily sees, the differential equation 

(5.4.1) (p xx + (p yy — k 2 (p = 0, 
the free surface condition 

G 2 

(5.4.2) cp y — mcp = 0, y = 0, m = — , 

g 



96 WATER WAVES 

and the condition at the bottom* 

(5.4.3) cp n = 0, y = — x tan eo. 

By introducing (as we have done before) the new dimensionless 
quantities x x = mx, y ± = my, o^ = a/ra, k x = k/m the conditions of 
the problem for cp(cc, y) can be put in the form 

(5.4.1)! <p xx + <p yy -k 2 (p = 0, ^ k ^ 1, 

(5.4.2) x (p y - <p = 0, y = 0, 

(5.4.3)! (p n = 0, y = — x tan a> 

after dropping subscripts. Since we require cp(x, y) to behave like 
e iccx e my = exp {ioc-LX-L + yi} at oo, it follows from (5.4.1) that 
- a 2 + m 2 - k 2 = and hence that a* + k\ = 1. Thus /b in (5.4.1)! 
(really it is k x ) is, as indicated, restricted to the range 5^ k ^ 1, 
and this fact is of importance in what follows.** Finally, we know 
from past experience that a singularity must be permitted at the 
origin. (In the problems treated earlier in this chapter we have 
prescribed only boundedness conditions at oo in a way which led to 
a statement concerning the uniqueness of the solution. In the present 
case we do not obtain a similar uniqueness theorem — in fact, as has 
been pointed out by Ursell [U.7, 8], Stokes showed that there exist 
motions different from the state of rest and which die out at oo. 
For these motions, however, the quantity k is larger than unity). 
We seek functions cp(x, y) satisfying the above conditions as the 
real or the imaginary part of a complex function f(z, z) which is 
analytic in each of the variables z = x + iy and its conjugate 
z = x — iy. In the two-dimensional cases, it was sufficient to consider 
analytic functions f(z) of one complex variable, but in the present 
case it is necessary to take more general functions since <p(x, y) is 
not a harmonic function. Note that we now use the variable z in a 
different sense than above, where it is one of the space variables; 
no confusion should result since the space variable z hardly occurs 
again in the discussion to follow. It is useful to calculate some of the 
derivatives of such functions with respect to x and y; we have, 
clearly: 

* Peters [P.6] solves the problem when the condition (5.4.3) is replaced by 
the more general mixed boundary condition q> n -f- ay = 0, a = const. 

** Involved in this remark is the assumption that the derivatives of the solution 
behave asymptotically the same as the derivatives of its asymptotic development ; 
but this is indeed the case, as we could verify on the basis of our final represen- 
tation of the solution. 






WAVES ON SLOPING BEACHES AND PAST OBSTACLES 97 

1 x = Jz z x I fz%x == Iz T/j) 

/, = *U. ~ h), 

7 XX I Jyy ^Jzz ' 

Consequently our differential equation (5.4.1)! can be replaced by 
the differential equation 

4 

since the real or the imaginary part of any solution of it is clearly 
a solution of (5.4.1 \. 

Among the solutions of the last equation are the following simple 
special solutions (obtained, for example, by separating the variables 
in writing / = j x {z) • / 2 (z)): 

f(z, z) = e^ z + 4 c , C = const., 
which, when J = — i for example, is of the form 

exp j 1 1 -f- — ) y J • exp { — i J 1 — — ) x \ , and this is a solution of 

(5.4.1 ) x which has the proper behavior at go, at least. (Actually, 
when combined with the factor e ikz , with z once more the space 
variable, the result is a harmonic function yielding a plane wave in 
water of infinite depth and satisfying the free surface condition). 
One can obtain a great many more solutions by multiplying the 
above special solution by an analytic function g(£) and integrating 
along a path P in the complex £-plane: 

(5.4.4) /(«,*)= _Lf ** + T !■*(£)#. 

2th J P 

By appropriate choices of the analytic function g(f ) and the path P, 
we might hope to satisfy the boundary conditions and the condition 
at go. This does, indeed, turn out to be the case. 

Still another way to motivate taking (5.4.4) as the starting point 
of our investigation is the following. It would seem reasonable to 
look for solutions of (5.4.1) in the form of the exponential functions 
(p = exp {mx -\- ly). However, since we wish to work with analytic 
functions of complex variables it would also seem reasonable to express 
x and y in terms of z = x + iy and z = x — iy, and this would lead to 

(p = exp J m I- - J — li J J | . In order that this function 



98 



WATER WAVES 



(which is clearly analytic in z and z separately) be a solution of 
(5.4.1 ) 1 we must require that m 2 + I 2 — k 2 = 0, and this leads at 

(k 2 z \ 
Cz + J , with f an ar- 
4 CJ 
bitrary parameter, as one can readily verify. The method used by 
Peters [P. 6] to arrive at a representation of the form (5.4.4) is better 
motivated though perhaps more complicated, since he operates with 
(5.4.1) in polar coordinates, applies the Laplace transform with 
respect to the radius vector, transforms the resulting equation to 
the Laplace equation, and eventually arrives at (5.4.4). 

One of the paths of integration used later on is indicated in Figure 
5.4.2. The essential properties of this parth are: it is symmetrical 
with respect to the real axis, goes to infinity in the negative direction 




£- plane 



Fig. 5.4.2. The path P in the £-plane 



of the real axis, enters the origin tangentially to the real axis and 
from the left, and contains in the region lying to the left of it a 
number of poles of g(£). (The path is assumed to enter the origin 
in the manner indicated so that the term z/C in the exponential 
factor will not make the integral diverge). Our discussion will take 
the following course: We shall assume g(£) to be defined in the £-plane 
slit along the negative real axis (and also on occasion on a Riemann 
surface obtained by continuing analytically over the slit). The choice 
of the symmetrical path P leads to a functional equation for g(J) 
through use of the boundary conditions (5.4.2) x and (5.4.3) 1? and vice 
versa a solution g(£) of the functional equation leads to a function 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



99 



(p(x } y) = 0te f(z, z) satisfying the boundary conditions. (By the 
symbols Jm and 0te we mean, of course, that the imaginary, or real, 
part of what follows is to be taken. ) We seek a solution of the functional 
equation which is defined and regular in the slit £-plane, with at 
most poles in the left half-plane (including certain first order poles 
on the negative imaginary axis), and dying out at oo like 1/f. Once 
such a function has been found, the prescribed conditions at oo will 
be seen to follow by deforming the path P over the poles into a path 
on the two edges of the slit along the negative real axis: the residues 
at the poles on the negative imaginary axis clearly would yield con- 
tributions of the type 

I k 2 z \ 
g(—ri) - exp J — irz + — ; — }> r > 0, which are easily seen to be 



4 (— ir) 

of the desired type at oo, while the remaining poles and the integral 
over the deformed path will be found to yield contributions that tend 
to zero when 0te z -> + oo. 

We begin this program by expressing the boundary conditions 
(5A.2) 1 and (5.4.3)! in terms of the function f(z, z). The first of 
these conditions will be satisfied if the following condition holds: 

(5.4.2)j <#w(f z — h + */) = °> z real > positive, 

as one readily sees. The condition (5.4.3 ) ± will be satisfied if 
n • grad op = 0, with n the unit normal at the bottom surface, i.e. 
if £%e {n • grad /} = 0, and the latter is given by 

®e {(/. + h) sin a + *(/. ~ h) cos M ) = °> 
or finally, in the form 

(5.4.3 )[ Jm {f z e~ ico — f-e ico } = 0, z = re~ i0i , r > 0. 

Upon making use of (5.4.4) in (5A.2){ the result is 

k 2 



(5.4.5) Jm 



-■f 

2m J j 



e **• T 4C 



C 



K 



+ i 



while (5.4.3 )[ yields 



g(C)dC = 0, 

z real, positive, 



(5.4.6) Jm 



2m J t 



e £(a ^ 4c 



' . k 2 



g(C)dC = o, 



re' 



r > 0. 



To satisfy the boundary condition (5.4.5) it is sufficient to require 
that g(£) satisfies the condition 



100 



WATER WAVES 



(5.4.7) 



C 



4C 



+ i 



g(C) = 0, f real, positive. 



The proof is as follows: If (5.4.7) holds, then the integrand G(z, z, £) 
in (5.4.5) is real for real z and real positive £. Hence G takes on values 
G, G at conjugate points £, f which are themselves conjugate, by the 
Schwarz reflection principle. Since the path P is symmetrical, as 
shown in Figure 5.4.2, it follows that d£ takes on values at £, £ that 

are negative conjugates. Thus the integral ( l/2m) ( 6r d£ is real when 

2 is real and (5.4.7) holds. In considering next (5.4.6) we first introduce 
a new variable s = £e~ i(a to obtain for z = re~ im the condition, 
replacing (5.4.6): 

(5.4.8) Jm—[ e 
2m J P i 

Here P' is the path obtained by rotating P (and the slit in the 
£-plane as well, of course) clockwise about the origin through the angle 
co. If g behaves properly at oo, and if the rotation of P' can be 
accomplished without passing over any poles of the integrand, we 
may deform P' back to P and obtain 



k 2 r 




k* 


4s • 


s - 


49 



g(se im ) e ia> ds = 0, r real. 



(5.4.8)' 



2m J j 



rs + 



k 2 r r 



k 2 

47 



g(se iU) ) e ia) ds = 0, r real. 



By the same argument as before we now see that the condition 
(5.4.6) will be satisfied provided that g(£) satisfies the condition 



(5.4.9) 



Jm g(£e ia) ) e iuy = 0, J real, positive. 



Thus if the function g(£) satisfies the conditions (5.4.7) and 
(5.4.9), the function f(z, z) constructed by its aid will satisfy the 
boundary conditions. As we have already remarked, g(£) must 
satisfy still other conditions —at oo, for example. In addition, we 
know from earlier discussions in this and the preceding chapter that 
it is necessary to find two solutions (p(x, y) and cp^x, y) of our problem 
which are "out of phase at oo", in order that a linear combination 
of them with appropriate time factors will lead to a solution having 
the form of an arbitrary progressing wave at oo. In this connection 
we observe that if the path P 1 of integration (as shown in Figure 
5.4.3) is taken instead of the path P (it differs from P only in reversal 
of direction of the portion in the upper half-plane), and if we define 
cp^x, y) as the imaginary part of / x (z, z) instead of its real part: 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



101 






(5.4.10) cp^x, y) = Jm—\ G(z, z, £)d£ = Jm f ± (z, a), 

2mJ Pi 

with G the same integrand as before, then (p x (x, y), by the same 
argument as above, will satisfy the boundary conditions provided 
that the function g(f ) also in this case satisfies the conditions (5.4.7) 




£- plane 



Fig. 5.4.3. The path P x in the f-plane 

and (5.4.9). It seems reasonable to expect that the integral over P x 
will behave the same as the integral over P when 0te z is large and 
positive (since the poles in the lower half-plane alone determine this 
behavior and the paths P and P x differ only in the upper half- 
plane) except that a factor i will appear, and hence that op and cp x 
will differ in phase at + °° ( m the variable x, that is) by 90°. This 
does indeed turn out to be the case. 

Thus to satisfy the boundary conditions for both types of standing 
wave solutions we have only to find a function g(£) satisfying the con- 
ditions (5.4.7) and (5.4.9) which behaves properly at oo— the last 
condition being needed in order that the path of integration can be 
rotated in the manner specified in deriving (5.4.8)'. To this end we 
derive a functional equation for g(f) by making use of these con- 
ditions. From (5.4.7) we have, clearly: 



¥ 



(5.4.11) If - ; + i\ g(C) = U- — -i\ g(C), C real, positive, 
while from (5.4.9) we have 



102 WATER WAVES 

(5.4.12) g(£)e- iaJ = g(& 2i<0 )e i(0 , f real, positive, 

both by virtue of the reflection principle. Eliminating g(£) from the 
two equations we obtain 



■).. 



(5.4.13) (f -^+i)g(C) =«*-(:- g -<)«(* 

This functional equation was derived for £ real and positive, but 
since g(f ) is analytic it is clear that it holds throughout the domain 
of regularity of g(£); it is the basic functional equation for g(f), a 
solution of which will yield the solution of our problem. Of course, 
this equation is only a necessary condition that must be fulfilled if 
the boundary conditions are satisfied; later on we shall show that the 
solution of it we choose also satisfies the condition (5.4.11), and hence 
the condition (5.4.12) will also be satisfied since (5.4.13) holds. 

We proceed now to find a solution g(f) of (5.4.13) which has all of 
the desired properties needed to identify (5.4.4) and (5.4.10) as 
functions furnishing the solution of our problem, as has been done by 
Peters in the paper cited above. 

We therefore proceed to treat the functional equation (5.4.13), 
which is easily put in the form: 

(5.4.14) gjg»_Y*..P+g-T 



4 

= e -2ia> (C + irM + ir 2 ) 
(C - «>i)(C - ir 2 ) 

with r x 2 



i ± Vi 



The numbers r x 2 are real since we know that k lies between and 1. 
It is convenient to set 

(5.4.15) iff, = W» , 

(C + ir^iC + ^r 2 ) 

in which h(£ ), like g(f ), is defined in the C-plane slit along the negative 
real axis. The function h(£) will have poles in the left half-plane, but 
only the poles at J = — ir 1 and J = — i> 2 of g(£) will be found to 
contribute a non-vanishing residue of f(z) for 0te z -> + oo, and 
this in turn would guarantee that f(z) behaves at oo on the free 
surface like Ae~ iz . For h(£) we have from (5.4.14) and (5.4.15) the 
equation 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 103 

[ ' A(f) (C-friMC-*-,) 

This equation is solved by introducing the function 1(C) by 

(5.4.17) logfc(f) = *(£)> 

and one finds at once that Z(f ) satisfies the difference equation 

(5.4.18) l(#*t) - 1(C) = log m(£) = w(f ). 

In solving this equation we shall begin by producing a solution Z(f ) 
free of singularities in the sector — co ^ arg f ^ o>, after which the 
function /*(£)— which is (cf. (5.4.17)) then also regular in the same 
sector — can be continued analytically into the whole £-plane slit 
along the negative real axis (or, if desired, into a Riemann surface 
having the origin as its only branch point) by using (5.4.16). As an 
aid in solving equation (5.4.18) we set 

co = out, < a ^ 1, 



(5.4.19 

= T a , Z(r a ) = L(r), w(t«) = W(t), 

and operate now in a r-plane. One observes that the sector — co 
< arg f < co in the £-plane corresponds to the r-plane slit along 
its negative real axis. For L(r) one then finds at once from (5.4.18) 
the equation 

(5.4.20) L(re 2ni ) - L(r) = W(r). 

Our object in putting the functional equation into this form 
(following Peters) is that a solution is now readily found by making 
use of the Cauchy integral formula. Let us assume for this purpose 
that L(r) is an analytic function in the closed r-plane slit along its 
negative real axis* (which would imply that 1(C) is regular in 
the sector — co ^ arg J ^ co, as we see from (5.4.19)); in such a 
case L(t) can be represented by the Cauchy integral formula: 

(5.4.21) L(r)= -L(t P^-di, 

with C the path in the £-plane indicated by Figure 5.4.4. If we 
suppose in addition that L(£) dies out at least as rapidly as, say, 
1/1 at oo, it is clear that we can let the radius R of the circular 
part of C tend to infinity, draw the path of integration into the two 
edges of the slit and, in the limit, find for L(t) the representation 

* We shall actually produce such a regular solution shortly. 



104 WATER WAVES 




£-plane 



Fig. 5.4.4. Path C in the £-plane 

If If 

(5.4.22) L( T ) = _ + , 

2m J 2m . 

in readily understandable notation. On making use of (5.4.20), and 
drawing the two integrals together, it is readily seen that L(r) is 
given by 

(5.4.23) L(t) = -L f ^1 ft. 

The path of integration is the negative real |-axis, and W(I~) is to 
be evaluated for arg £ = — n. Since W(£) has no singularities (cf. 
(5.4.18)), it follows that L(r) as given by (5.4.23) is indeed regular 
in the slit r-plane. L(r) also has no singularity on the slit except 
at the origin, where it has a logarithmic singularity. Since the 
numerator in the integrand behaves like l/| a , a > 0, at oo (cf. 
(5.4.19), (5.4.18), (5.4.16)), it is clear that the function L(t) dies 
out like 1/r at oo in the r-plane. This function therefore has all of 
the properties postulated in deriving (5.4.23) from (5.4.21), and 
hence is a solution of the difference equation (5.4.20) in the slit 
plane including the lower edge of the slit. 

A solution of (5.4.18) can now be written down through use of 
(5.4.19); the result is: 

1 f° Wra(£ a ) 

(5.4.24) m - -±V>«, 



with m(£ a ) to be evaluated for arg £ = - — n. This solution is valid 
so far only for f in the sector — co ^ arg { ^ co, where it is regular, 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 105 

as we know from the discussion above. However, it is necessary to 
define the function h(£) = e l(Z) (cf. (5.4.17)) in the entire slit f- 
plane, and this can be done by analytic continuation with the aid 
of the functional equation (5.4.16). In the process of analytic con- 
tinuation, starting with the original sector in which Z(£), and hence 
h(£), is free of singularities, one sees that the only singularities which 
could occur in continuing into the upper half-plane, say, would arise 
from the function on the right hand side of the equation (5.4.16). 
The only singularities of this function occur obviously at f = ir 1 2 . 
Consequently no singularity of h(£) appears in the analytic con- 
tinuation into the upper half-plane, through widening of the sector 
in which h(£) is defined, until the points J = ir 1 and f = ^Y 2 have 
been covered, and one sees readily from (5.4.16) that the first such 
singularities of h(£) — -poles of first order — appear at the points 
r 12 exp {i(2co + tz/2)} 9 the next at r 1>2 exp {i(4ico + nj2)}, etc., 
though some, or all, of these poles may not appear on the first sheet 
of the slit £-plane, depending on the value of the angle <x>. The con- 
tinuation into the lower half-plane is accomplished by writing 
(5.4.16) in the equivalent form 

h(Ce- 2iC0 ) 1 



(5.4.16)' 



h(C) m(£e- 2ito ) 



Again we see that poles will occur in the lower half-plane in the 
course of the analytic continuation, this time at r 1 2 exp { — ?'(2co+7z;/2} 
r 12 exp {— i(4*co -f tt/2)}, etc. The situation is indicated in Figure 
5.4.5; h(£) lacks the singularities of g(f ) at the points — h\ and 
— ir 2 (cf. (5.4.15)). Thus the function h(£) is defined in the slit f- 
plane. (It can also be continued analytically over the slit which 
permits a rotation of the path of integration.) We see that h(£) 
may have poles in the open left half-plane, on two circles of radii 
r 1 and r 2 , but the poles closest to the imaginary axis are at the 
angular distance 2co from it. There is also a simple pole of h(£) at 
the origin, but g(£) (cf. (5.4.15)) is regular there. 

The behavior of h(£) at oo in the slit plane is now easily discussed: 
In the original sector we know from (5.4.24) that Z(f ) dies out at oo 
like l/£ 1/oc . Hence h(£) = e l(!:) is bounded in the sector, and since the 
right hand side of (5.4.16) is clearly bounded at oo it follows that 
h(£) is bounded at oo in the £-plane. 

The function g(f) = £/*(£)/(£ + ^)(C + ir 2 ) (cf. (5.4.15)) can now 
be seen to have all of the properties needed to identify the functions 



106 



WATER WAVES 



£ - plane 




Fig. 5.4.5. The singularities of h(£) and g(£) 

f(z) in (5.4.4) and f^z) in (5.4.10) as functions whose real part and 
imaginary part, respectively, yield the desired standing wave solutions 
of our problem. To this end we write down the integrals 

e ;* + !g\ »(C) 

2m ' 



(5.4.25) /,/, 






4C 



# 



over the paths indicated in Figure 5.4.6, where the direction is in- 
dicated only on the part of the path in the lower half-plane, since 
the paths P, P 1 differ only in the direction in which the remainder 
of the path is traversed. 

Since h(£) is bounded at oo, and 0te z > 0, the integrals clearly 
converge. One sees also that the paths of integration can be rotated 
through the angle co about the origin without passing over singularities 
of the integrand, and also without changing the value of the in- 
tegrals. (This was needed in deriving (5.4.8)'.) We prove next that 
g(£) satisfies the boundary condition (5.4.11). To begin with, we shall 
show that 1(C) as defined by (5.4.24) is real when f is real and positive. 
Once this is admitted to be true, then h(£) as given by (5.4.17) would 
have the same property, and the function g(£) defined by (5.4.15) 
would easily be seen to satisfy the condition (5.4.11). We have, then, 
only to show that Z(£) is real for real £, and this can be seen as follows: 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 107 



£ - plane 




Fig. 5.4.6. The paths P, P x in the £-plane 



In (5.4.24) log ?n(£ a ) is to be evaluated for arg £ = — n. But in this 
case one sees easily from the equation (5.4.16) defining m(£ a ) (with 
a = oJj'ti, cf. (5.4.19)) that ra(£ a ) has its values on the unit circle 
when arg £ = — jz, and hence its logarithm is pure imaginary on 
the path of integration; it follows at once from (5.4.24) that /(£) 
is real for f real and positive. Since g(f ) was constructed in such a 
way as to satisfy (5.4.13) we know that (5.4.12) is satisfied auto- 
matically. Thus our standing wave solutions satisfy the boundary 
conditions. 

Finally, we observe that the behavior of / and f ± for £%e z -» oo 
is what was prescribed. To this end we deform the path of integration 
into a path running along the two banks of the slitted negative 



real axis. The residues at f 



ir 



1,2 



contribute terms already 



discussed above which furnish the desired behavior for £%e z -> + °°« 
We have, then, only to make sure that the residues at the remaining 
poles and the integrals along the slit make contributions which die 
out as 0te z -> + oo. As for the residues at the poles at the points 



108 



WATER WAVES 



Cn = r i 2 ex P (± i(2nco + n/2)}, n = 1,2, 



/5 . ., we observe that these 

contributions are of the form Ae z ^n, but since — co ^ arg z ^ it 
is clear that these contributions die out exponentially when z tends 
to infinity in the sector — co ^ arg z ^ 0. As for the integrals along 
the slit, they are known to die out like 1/z, as we have seen in similar 
cases before, or as one can verify by integration by parts. Thus all 
of the conditions imposed on f(z) and f ± (z) are seen to be satisfied. 
We observe, however, that the integrals in (5.4.25) over the paths 
P and P 1 converge only if 0te z ^ 0, and hence this representation 
of our solution is valid only if the bottom slopes down at an angle 
^ jz/2. For an overhanging cliff, when co > nj2, the solution can be 
obtained by first swinging the path of integration clockwise through 
90° (and swinging the slit also, of course); the resulting integrals 
would then be valid for all z such that J>m z ^ and the solutions 
would hold for < co ^ n. 

It is perhaps of interest to bring the final formulas together for 
the simplest special case, i.e. the dock problem for two-dimensional 
motion (first solved by Friedrichs and Lewy [F.12]), in which the 



; 


y 


^^s^^ 


^-^ /— ^ /- 







Fig. 5.4.7. The dock problem 



angle co has the value n, as indicated in Figure 5.4.7.* In this case 
the function 1(C) is given by 

1 f° log 



(5.4.26) 






d£, 



and the integral defines it at once in the entire slit £-plane. The 
standing wave solutions (p(x, y) = &e f(z) and <p x (x, y) = J'm f^z) 
are determined through 

* As was mentioned in section 5.1, the dock problem in water of uniform 
finite depth and for the three-dimensional case was first solved by Heins [H.13] 
with the aid of the Wiener-Hopf technique. 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



109 



(5.4.27) 



(5.4.28; 



/(*: 



/i(«: 



C + i 



2m J j 



df, 



df, 



with h(C) defined by 

(5.4.29) 

As was remarked above 



>i(C) 



the integrals in (5.4.27) and (5.4.28) con- 
verge only if 0te z ^ 0. However, the analytic continuation into the 
entire lower half-plane is achieved simply by swinging the paths 
P and P x into the positive imaginary axis (which can be done since 
h(£) is bounded at oo), while staying on the Riemann surface of /&(£), 
and these integrals are then valid for all z in the lower half-plane. 
Finally, it is also of interest to remark that the functions f(z) 
and f x (z) do not behave in the same way at the origin: the first is 
bounded there, and the second is not, and this behavior holds not 
only for the special case of the dock problem, but also in all cases 
under consideration here. 



5.5. Diffraction of waves around a vertical wedge. 
Sommerfeld's diffraction problem 

In this section we are primarily concerned with the problem of 
determining the effect of a barrier in the form of a vertical rigid 
wedge, as indicated in Fig. 5.5.1, on a plane simple harmonic wave 




Fig. 5.5.1. Diffraction of a plane wave by a vertical wedge 

coming from infinity. In this case it is convenient to make use of 
cylindrical coordinates (r, 6, y). We seek a harmonic function 



110 WATER WAVES . 

0(r, 6, y; t) in the region <0 <v, — h < y < 0, i.e. in the region 
exterior to the wedge of angle 2n — v and in water of finite depth h 
when at rest. The problem is reduced to one in the two independent 
variables (r, 6) by setting 

(5.5.1) 0(r, 6, y; t) = f(r, 6) cosh m(y + h)e iat . 

The boundary conditions d = for 6 = 0, 6 = v corresponding 
to the rigid walls of the wedge yield for f(r, 6) the boundary con- 
ditions 

(5.5.2) f e = 0, 6 = 0, 6 = v. 

The free surface condition g@ y + ^tt = at y = yields the con- 
dition 

(5.5.3) m tanh mh = o 2 /g, 

while the condition y = at the bottom y = — h is satisfied 
automatically. Once any real value for the frequency o is prescribed, 
equation (5.5.3) is used to determine the real constant m — which 
will turn out to be the wave number of the waves at oo — , and we 
note that (5.5.3) has exactly one real solution of m except for sign; 
if the water is infinitely deep we have m = o 2 /g, and the function 
cosh m(y -f- h) in (5.5.1) is replaced by e my . 

Thus the function f(r, 6) is to be determined as a solution of the 
reduced wave equation 

(5.5.4) Vf r>0) / + m 2 / = 0, 0<r<oo, <6 <v, 

subject to the boundary conditions (5.5.2). Actually, we shall in the 
end carry out the solution in detail only for the case of a reflecting 
rigid plane strip (i.e. for the special case v = 2jz), but it will be seen 
that the same method would furnish the result for any wedge. It is 
convenient to introduce a new independent variable q, replacing r, 
by the equation r = g/m; in this variable equation (5.5.4) has the 
form 

(5.5.5) V% td] f+f = 0, 0<g<co, <6 <v, 

and we assume this equation as the basis for the discussion to follow. 
So far we have not formulated conditions at oo, except for the 
vague statement that we want to consider the effect of our wedge- 
shaped barrier on an incoming plane wave from infinity. Of course, 
we then expect a reflected wave from the barrier and also diffraction 
effects from the sharp corner at the origin. In conformity with our 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 111 

general practice we wish to formulate these conditions at oo in such 
a way that the solution of the problem will be uniquely determined. 
It has some point to consider the question of reasonable conditions 
at oo which determine unique solutions of the reduced wave equation 
under more general circumstances than those considered in the 
physical problem formulated above. For general domains it is not 
known how to formulate these conditions at oo, and, in fact, it would 
seem to be a very difficult task to do so since such a formulation 
would almost certainly require consideration of many special cases. 
In one special case, however, the appropriate condition to be im- 
posed at infinity has been known for a long time. This is the case 
in which any reflecting or refracting obstacles lie in a bounded domain 
of the plane, or, stated otherwise, it is the case in which a full neigh- 
borhood of the point at infinity is made up entirely of the homogeneous 
medium in which the waves propagate. In this case, the condition 
at oo which determines the "secondary" waves uniquely is Sommer- 
feld's radiation condition, which states, roughly speaking, that these 
waves behave like a cylindrical outgoing progressing wave at oo. 
However, if the reflecting or refracting obstacles extend to infinity, 
the Sommerfeld condition may not be appropriate at all. Consider, 
for example, the case in which the entire ^r-axis is a reflecting barrier 
(i.e. the case v = jz), and the primary wave is an incoming plane wave 
from infinity. It is clear on physical grounds that the secondary wave 
will be the reflected plane wave, which certainly does not behave 
at oo like a cylindrical wave since, for example, its amplitude does 
not even tend to zero at oo. Another case is that of Sommerfeld 's 
classical diffraction problem in which an incoming plane wave is 
reflected from a barrier consisting of the positive half of the ^-axis. 
In this case, the secondary wave has both a reflected component 
which has a non-zero amplitude at oo, and a diffracted part which 
dies out at oo. A uniqueness theorem has been derived by Peters and 
Stoker [P-19] which includes these special cases; we proceed to give 
this proof both for its own sake and also because it points the way 
to a straightforward and elementary solution of the special problem 
formulated above. In Chapters 6 and 7 a different way of looking 
at the problem of determining appropriate radiation conditions is 
proposed; it involves considering simple harmonic waves (Chapter 6), 
or steady waves (Chapter 7) as limits when t -> oo in appropriately 
formulated initial value problems which correspond to unsteady 
motions. 



112 



WATER WAVES 



The uniqueness theorem, which is general enough to include the 
problem above, is formulated in the following way: We assume that 
f(x, y) is a complex- valued solution* of the equation 

(5.5.6) V 2 /+/ = 

in a domain D with boundary r, part of which may extend to in- 
finity. It is supposed that any circle C in the x, y-plane cuts out of 
D a domain in which the application of Green's formula is legitimate, 
and, in addition, that the boundary curve r outside a sufficiently 




Fig. 5.5.2. The domain D 



large circle consists of a single half-ray R going to oo (cf. Figure 
5.5.2).** On the boundary r the condition 

(5.5.7) f n = 

is imposed, i.e. the normal derivative of / vanishes, corresponding 
to a reflecting barrier. (We could also replace this condition on part, 
or all, of r by the condition / = 0.) We now write the solutions of 
f in D which satisfy (5.5.6) in the form 

(5.5.8) f = g+K 

in order to formulate the conditions at oo in a convenient way. 
What we have in mind is to separate the solution into a part h 
which satisfies a radiation condition and a part g which contains, 

* It is natural to consider such complex solutions, since, for example, a plane 
wave is obtained by taking f(g, 6) = exp { io cos (6 + a)}. 

** Our theorem also holds if D is the more general domain in which the ray R 
is replaced at oo by a sector, and the uniqueness proof given below holds with 
insignificant modifications for this case also. 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 113 

roughly speaking, the prescribed incoming wave together with any 
secondary reflected or refracted waves which also do not satisfy a 
radiation condition. More precisely, we require h to satisfy the 
following radiation condition: 

r ?h 2 

(5.5.9) lim 

£)— *-00 J C 

Here C is taken to be a circle, with its center (cf. Figure 5.5.2) 
on the ray R going to infinity, and with radius q so large that all 
obstacle curves except a part of R lie in its interior. This condition 
clearly follows from the well-known Sommerfeld radiation condition, 
which requires that 



dq 



ds = 0. 



(5.5.9)! lim gi 



e + «) 



-»o 



uniformly in 6, and, incidentally, this is a condition independent of 
the particular point from which g is measured; we observe that if h 
behaves at oo like e^ iQ l\Zg, i.e. like an outgoing cylindrical wave, 
then condition (5.5.9) 1 is satisfied. We shall make use of the radiation 
condition in the form (5.5.9) in much the same way as F. John [J.5] 
who used it to obtain uniqueness theorems for (5.5.6) in cases other 
than those treated here; his methods were in turn modeled on those 
of Rellich [R.7]. 

The behavior of the function g at infinity is prescribed as follows: 

(5.5.10) g~gi +g 2 at °°> 

with g ± a function that is once for all prescribed,* while g 2 is a function 
satisfying the same radiation condition as h, i.e. the condition 
(5.5.9). (That the behavior of g at oo is fixed only within an additive 
function satisfying the radiation condition is natural and inevitable. ) 
Finally, we prescribe regularity conditions at re-entrant points 
(such as A, B, C in Figure 5.5.2) of the boundary of D; these con- 
ditions are that 

(5.5.11) f(Q,d)~c l9 f Q {Q,0)~% k<l, 

Q k 

with (q, d) polar coordinates centered at the particular singular point, 
and c 1 and c 2 constants. (These conditions on / mean physically that 
the radial velocity component may be infinite at a corner, but not 

* How the function g x should be chosen is a matter for later discussion. 



114 WATER WAVES 

as strongly as it would be for a source or sink.) At other boundary 
points we require continuity of / and its normal derivative. 
We can now state our theorem as follows: 

Uniqueness theorem: A solution / of (5.5.6) in D is uniquely determined 
if it 1) satisfies the boundary condition (5.5.9); 2) admits of a decom- 
position of the form (5.5.8) with h a function satisfying (5.5.9), g a 
function behaving as prescribed by (5.5.10) at oo; and 3) satisfies the 
regularity conditions at the boundary of D. 

The proof of this theorem will be given shortly, but we proceed 
to discuss its implications here. The theorem is at first sight somewhat 
unsatisfactory since it involves the assumption that every solution 
considered can be decomposed according to (5.5.8), with g(g, 6) a 
certain function the behavior of which at oo, in so far as the leading 
term g x (cf. (5.5.10)) in its asymptotic development is concerned, is 
not given a priori. However, it is not difficult in some instances at 
least to guess, on the basis of physical arguments, how the function 
§i(Q> 0) should be defined. For example, suppose the domain D 
consisted of the exterior of bounded obstacles only. In such a case it 
seems clear that g^Q, 6) should be defined as the function describing 
the incoming wave — either as a plane wave from infinity, say, or a 
wave originating from an oscillatory source — since bounded obstacles 
give rise only to reflected and diffracted components which die out 
at oo and which could be expected to satisfy the radiation condition. 
Even if there is a ray in the boundary that goes to oo (as was postulated 
above), it still would seem appropriate to take g^Q, 6) as the function 
describing the incoming wave, provided that it arises from an oscil- 
latory point source,* since such a source would hardly lead to reflcted 
or refracted secondary waves that would violate the radiation con- 
dition. However, if the incoming wave is a plane wave and an 
obstacle extends to oo, one expects an outgoing reflected wave to 
occur which would in general not satisfy the radiation condition; 
in this case the function g x (o, 6) should be taken as the sum of the 
incoming plane wave and an outgoing reflected wave. For example, 
one might consider the case in which the entire ,r-axis is a reflecting 
barrier, as in Figure 5.5.3. In this case one would in an altogether 
natural way define g^Q, 6) as the sum of the incoming and of the 
reflected wave as follows: 

* The same statement would doubtlessly hold if the disturbance originated 
in a bounded region, since this case could be treated by making use of a distribu- 
tion of oscillatory point sources. 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 115 

(5.5.12) g^g, 0) = e ie cos (d ~ a) + e iQ cos {d+a) , 

with a the angle of incidence of the incoming plane wave. If we were 
then to set / = g x + h (i.e. we set g = g 1 everywhere) and prescribe 
that h should satisfy the radiation condition, it is clear that we would 




Fig. 5.5.3. Infinite straight line barrier 

have a unique solution by taking h = 0. Our uniqueness theorem 
does not apply directly here since there are two infinite reflecting 
rays going to oo, but it could be easily modified so that it would 
apply to this case. Thus we have — for the first time, it seems — a 



/ 9--TT + Q 

Fig. 5.5.4. Sommerf eld's diffraction problem 

uniqueness theorem for this particularly simple problem of the 
reflection of a plane wave by a rigid plane. A less trivial example is 
the classical Sommerfeld diffraction problem — in effect, a special 



116 WATER WAVES 

case of the problem with which our present discussion began — in 
which a plane wave coming from infinity at angle a to the #-axis is 
reflected and diffracted by a rigid half-plane barrier along the 
positive #-axis, as indicated in Figure 5.5.4. In this case it seems 
plausible to define the function g x (Q, 6) as follows: 

/ e i Q cos (0-oc) _j_ e ie cos (0+a) ? < fl < n _ a 

(5.5.13) g 1 {Q,6)= le i6C0S(d - a \ 7r-a<0<7r+a 

I 0, n + a < 6 < 2n. 

This function is, of course, discontinuous, corresponding to the 
division of the plane into the regions in which a) the incoming wave 
and its reflection from the barrier coexist, b) the region in which 
only the wave transmitted past the edge of the barrier exists, and 
c) the region in the shadow created by the barrier. Again we would 
be inclined to take g = g x (cf. (5.5.8) and (5.5.10)) and set / = g x -f- h, 
with h satisfying the radiation condition. Of course, the function 
h(g, d) in (5.5.8) representing the diffracted wave would then also 
be discontinuous in that case since the sum g x -f- h is everywhere 
continuous. It will be seen that the well-known solution given by 
Sommerfeld can be decomposed in this way and that h then satisfies 
the radiation condition. Our uniqueness theorem will thus be shown 
to be applicable in at least the important special case of particular 
interest in this section. 

One might hazard a guess regarding the right way to determine 
the function g in all cases involving unbounded domains: it seems 
highly plausible that it would always be correctly given by the 
methods of geometrical optics. By this we mean, from the mathemati- 
cal point of view, that g would be the lowest order term in an asymp- 
totic expansion of the solution / with respect to the frequency of the 
motion that is valid for large frequencies; the methods of geometrical 
optics would thus be available for determining g. However, to prove 
a theorem of such generality would seem to be a very difficult task 
since it would probably require some sort of representation for the 
solution of wave propagation problems when more or less arbitrary 
domains and boundary data are prescribed. 

Once having proved that the solution of Sommerfeld's diffraction 
problem could be decomposed in the way indicated above into the 
sum of two discontinuous functions, one of which satisfies the 
radiation condition, it was observed that the latter fact opens the 
way to a new solution of the diffraction problem which is entirely 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 117 

elementary, straightforward, and which can be written down in a few 
lines. In other words, once the reluctance to work with discontinuous 
functions is overcome, the solution of the problem is reduced to 
something quite elementary by comparison with other methods of 
solution. The problem was solved long ago by Sommerfeld [S.12], 
and afterwards by many others, including Macdonald [M.l], Bateman 
[B.5], Copson [C.4], Schwinger [S.5], and Karp [K.3]. 

We shall first prove the uniqueness theorem. Afterwards, the simple 
solution of Sommerfeld 's problem just referred to will be derived; 
this solution is in the form of a Fourier series. The Fourier series 
solution is next transformed to furnish a variety of solutions given 
by integral representations, including the familiar representation 
given by Sommerfeld. The new representations are particularly 
convenient for the purpose of discussing a number of properties of 
the solution. In particular, two such representations can be used to 
show that the function h in the decomposition / = g 1 -\- h (cf. 
(5.5.13)) satisfies the radiation condition, and that our solution / 
satisfies the regularity conditions at the origin; thus the solution is 
shown, by virtue of our uniqueness theorem, to be the only one which 
behaves at oo like g 1 plus a function satisfying the radiation condition. 
The Stokes' phenomenon encountered in crossing the lines of discon- 
tinuity of the functions g x and h is also discussed. 

The uniqueness theorem formulated above is proved in the following 
way. Suppose there were two solutions / and /* (cf. (5.5.8)) with 
/* given by 

(5.5.14) f*(Q,0)=g*(Q,0) +h*(Q,0). 

We introduce the difference %(q, Q) of these solutions: 

(5.5.15) *(M) = /(M)-/*(e>0) 

= g (Q, e) - g*( e , 6) + k( e , e) - h*( e , 6) 

and observe that %(q, 6) satisfies the radiation condition (5.5.9), by 
virtue of the Schwarz inequality, since h, h*, and the difference 
g — g* all satisfy it by hypothesis; thus we have 



lim I 

0— >-Q0 J C 






2 

ds = 0. 



(5.5.16) 

] c 

The complex-valued function # is decomposed into its real and 

imaginary parts: 

(5.5.17) X = Xi + hv 

and Green's formula 



118 



WATER WAVES 



(5.5.18) 



jy (*« v '*i 



X^X^dx dy 



I 



%2 



hi 

dn 



a 



n) 



ds 



is applied to X\ an d X2 m the domain D* indicated in Figure 5.5.5. 
The domain D* is bounded by a circle C so large as to include all of 
the obstacles in its interior except R, by curves which exclude the 




Fig. 5.5.5. The domain D* 

prolongation of R into the interior of C, and by curves excluding 
the other bounded obstacles. By (5.5.15), % is a solution of (5.5.6) 
which also clearly satisfies the boundary condition (5.5.7). Since 
V 2 Xi = — Xi anc * V 2 ^ 2 = — Xv ^ follows that the integrand of the 
left-hand side of (5.5.18) vanishes. Because of the regularity conditions 
at boundary points we are permitted to deform the boundary curve 
r* into the obstacle curves, and it then follows from the boundary 
condition (5.5.7) that 

(5-5-19) f c (X*Xi n -XiXO ds = 

since the contributions at the obstacles all vanish. We now make 

use of the easily verified identity 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 119 

(5.5.20) 2( X2Xln - X1X2J = \Xn\ 2 + \X\ 2 - \Xn + h? 
to deduce from (5.5.19) the condition 

(5.5.21) j c (\Xn\ 2 + Izl 2 )^ - \ c \Xn + iX?ds = 0, 

from which we obtain, in view of (5.5.16) and % n = d%\dg on C: 

(5.5.22) lim f | Z | 2 ds = lim f \ %n \* ds = §. 

From the boundary condition (5.5.7), as applied on R, we see that 
%(g, 6) can be continued as a periodic function of period 4>ti in 6 
on C; hence £ can be represented for all sufficiently large values of 
g by the Fourier series 

00 nd 

(5.5.23) X = Z A ni2(Q)c°s—, 

l 

with A n/2 (g), the Fourier coefficient, a certain linear combination 
of the Bessel functions J n/2 (g) and 7 n/2 (^), since % is a solution of 
(5.5.6). The Fourier coefficients are given by 

1 C 2n nd 

(5.5.24) A nl2 ( 9 ) = -\ z(e,0)cos — d0, 

and consequently we have 



(5.5.25) \Q*A nl2 (o)\ ^ 



q -[ n \x\dd 
n Jo 



< /^ lr! 2 ^ 



-r^| lxl2ds - 

It follows at once from (5.5.22) that the Fourier coefficients behave 
for large g as follows: 

(5.5.26) lim gi A n/2 (g) = 0. 

Q—>CG 

Since the Bessel functions J n j 2 (g) and Y n / 2 (g) all behave at oo like 
l/\/g, it follows that all of the coefficients A n j 2 (Q) must vanish. 
Consequently % vanishes identically outside a sufficiently large circle, 
hence it vanishes* throughout its domain of definition, and the 

* This could be proved in standard fashion since % is now seen to satisfy 
homogeneous boundarv conditions in the domain Z)* of Ficr. 5.5.5. 



120 WATER WAVES 

uniqueness theorem is proved. As was stated above, this uniqueness 
proof is much like that of Rellich [R.7]. 

The above proof can be modified easily in such a way as to apply 
to a region with a sector, rather than a ray, cut out at oo. The only 
difference is that the Fourier series for %(g, 6) would then not have 
the period 4tz; and that the Bessel functions involved would not be 
of index n/2. 

Once it has become clear that the decomposition of the solution 
into the sum of the two discontinuous functions g(g, 6) and h(g, 6) 
defined earlier is a procedure that is really natural and suitable for 
this problem, one is then led to the idea that such a decomposition 
might be explicitly used in such a way as to determine the solution 
of the original problem ((cf. (5.5.8)) in a direct and straightforward 
way. Our next purpose is to carry out such a procedure. 

We set (cf. Figure (5.5.4) and equation (5.5.13)): 

(5.5.27) f( Q ,0) =g(e,B) +h( Q ,d) 

with g(g, 6) defined by 

e ig cos (0-a) _|_ e iQ cos (0+a) j q < Q < n _ a 

^ cos(0 - a) , 7r-a<0<7r+<x 

0, n + a < 6 < 2tz. 



(5.5.28) g(Q,0) 



In addition we have 

(5.5.29) f = for 6 = 0, 6 = 2tz 



and we also require 

(5.5.30) lim -y/g (— + ih) = uniformly in 6, 



since the validity of the radiation condition in this strong form can 
be verified in the end. 

The desired solution will be found by developing /(£>, 6) into a 
Fourier series in 6 for fixed £>, and determining the coefficients of the 
series through use of the radiation condition in the strong form 
(5.5.30); afterwards, the series can easily be summed to yield a 
convenient integral representation of the solution. That such a 
process will be successful can be seen very easily: The Fourier series 
for f(g. 6) will, on account of the boundary condition (5.5.29) and the 
fact that / is a solution of the reduced wave equation, be of the form 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 121 

^c n J nj2 (g) cos nd/2; the Fourier coefficients for g(g, 6) as defined 
by (5.5.28) are given in terms of integrals of the form 



•-f 

Jo 



2n nO 

e iQ cos (0±«) CQS ™ dQ 

2 



since this function also satisfies the condition g e = for = 0, 2tt. 
Since J n / 2 (g), and its derivatives as well, behave like 1/^/q for large 
values of g and the integrals I n — by a straightforward application 
of the method of stationary phase, for example, — also behave in 
this way, it is clear that the limit relation (5.5.30) when used in 
connection with (5.5.27) will serve to determine the coefficients c n . 
We proceed to carry out this program. The finite Fourier transform 
/ of / is introduced by the formula 

C 2n nd 

(5.5.31) f(Q f n)=\ f(o, 6) cos — dd. 

Jo 2 

Since f e = for 6 = 0, 2ji we find for f ed the transform 

n 2 - 

(5.5.32) fee = - — f, 

4 

by using two integrations by parts. Since / is a solution of 

(5.5.33) g% g +gf Q + fee + g 2 f = 0, 
it follows that / is a solution of 

(5.5.34) g% e + gf Q + ^ _ ^ J = , 

and solutions of this equation are given by 

(5.5.35) f{Q,n) = a n Jni<i{g)- 

(The Bessel functions Y nj2 (g) of the second kind are not introduced 
because they are singular at the origin; the solution we want is in 
any case obtained without their use.) 

The transform of g(g, 6) is, of course, given by 

(5.5.36) g(g,n)=\ g( Q ,d) cos -dO, 

Jo 2 

and we have, in view of (5.5.8), the relation: 

' 2n . , ... nd .„ , . . r 2n 



C Zn nd f 2n 

(5.5.37) h(g, 6) cos - dd = a n J n/2 (g) - g( Q , 6) cos 

Jo 2 Jo 



dd 

Jo 2 J 2 

or, also : 



122 WATER WAVES 

(5.5.38) h(g, n) = a n J n , 2 (g) - g(g, n). 

We must next apply the operation \/g (d/dg -f* i) to both sides 
of (5.5.37) and then make the passage to the limit, with the result* 

nd 



dd 
2 



(5.5.39) = lim V? (^ +*) a n J nli ( Q ) - f g(g, 6) cos 

q^*> \dg / L Jo 

Since the functions J n ^(g) behave asymptotically as follows: 

J - /2(e) ~)^ cos ( e -i-f) 

and since these asymptotic expansions can be differentiated, we have 

(5.5.40) (L + i) J nl2 (g) ~ |/A ^(^/4-W4) 

as an easy calculation shows. The behavior of the integral over g 
can be found easily by the well-known method of stationary phase, 
which (cf. Ch. 6.8) states that 



Cyieyww do ~ 1/— ^ — vH«y(^ (a)± i) 

Ja f Q \<P"(<*)\ 

in which a is a simple zero of the derivative (p'(6) in the range 
a < 6 < b, and the ambiguous sign in the exponential is to be taken 
the same as the sign of 99" (a). In the present case, in which g(g, 6) 
is defined by (5.5.28) one sees at once that there are three points 
of stationary phase, i.e. at 6 = a, 6 = n — a, and 6 = n + a. Of 
the three contributions only the first, i.e. the contribution at 6 = a,** 
furnishes a non-vanishing contribution for g -> 00 when the operator 
Vg(d/dg -j- i) is applied to it; one finds, in fact: 



(5.5.41) h-+i)J £(M)cos- d0~2|T 



27r wa 

COS £' 



g 2 

Use of (5.5.40) and (5.5.41) in (5.5.39) furnishes, finally, the coef- 
ficients a n : 

TlOL . nn 

(5.5.42) a n = 2n cos — e*T- 

The Fourier series for f(g, 6) is 

* It should be noted that the argument goes through if the radiation condition 
is used in the weak form. 

** This has physical significance, since it says that only the incoming wave is 
effective in determining the Fourier coefficients of the solution. 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



123 



1 - 1 °° - nf) 

f(Q,0) = —f(Q>O)+- I t(Q,n) cos - 
2ji n ~i 2 

or, from (5.5.35) and (5.5.42), 

* inn fiOL fid 

f(Q> 0) = J<i(Q) + 2 I^ JnliiQ) COS — COS — ' 

It is not difficult to sum the series for f(g, 6). If we use the represen- 
tation (for a derivation, see Courant-Hilbert [C.10, p. 413]) 



(5.5.43) 



e ivn /» 

2m J t 



e-2^-c) C-'- 1 <%, 



where P is the path in the complex £-plane shown in Figure 5.5.6, 
we find that f(g, 6) can be expressed as the integral of the sum of 



£-plane 




Fig. 5.5.6. The path P in the £-plane 



a constant plus four geometric series. The summation of the geometric 
series and a little algebra yields, finally, a solution in the form 



(5.5.44) 
f(Q,0) 



1 C e ~o\^~d 
8m J P C 



1 if . Sn\ 1 i 

"1 i~f . Zn\ ' ~ 



«+»4" v 



s-.-iM: 



+ 






3tt\ 



£2+^2 



— - a— o — 



1 if Zn\ 

& - ^-2l a - -¥, 



d£. 



We proceed to analyze the solution (5.5.44) of our problem with 
respect to its behavior at oo and the origin, and we will show that 



124 



WATER WAVES 



the conditions needed for the validity of the uniqueness theorem 
proved above are satisfied. We will also transform it into the solution 
given by Sommerfeld (cf. equation (5.5.47)). Not all of the details 
of these calculations will be given: they can be found in the paper 
by Peters and Stoker [P.19]. 
If we set 



(5.5.45) f( Q ,Q) =I(q,0 + 


■a) +/(e,0- a ) 


and define I(g, x) by 


(5.5.46) I(q, x) = 1 e~2 V"V 

8m J P 


- 1 if Sn\ 1 if 3n\ 

£2 + e2\ x+ J) ^2 + ^r" + ¥j 

1 if 3n\ ■"• 1 if 3n\ 



<K 



we see on comparison with (5.5.44) that (5.5.45) defines f(g, 6) 
correctly as the solution we wish to investigate. 

Let us first obtain the solution in the form given by Sommerfeld. 
To this end, the denominators of the fractions in square brackets 
in (5.5.46) are rationalized, and the fractions combined to yield 






I(Q, M) = 



,-sh: 



c 



3n 
i— 



3n 

i—- X 



C 2 + 2£2 e 4 cos f- + 2?'C2 e 4 cos -f- + 1 
C 2 + 2i£ cos x - 1 



dC. 



One can then verify readily that I satisfies the differential equation 



dl , . 
■ 2—+(2icosx)I 



-I" 

4>m J 2 



e 2 



H) 



1 .371 -, 3 .3ti «. 

l+2C"2^Tcos- +2^~2g*Tcos- +;~ 2 



371 

i — /c 

£ 4 COS - 

2 

2jri 



g 2 



•H) ! 



(C 2 + »f 2 ) d£. 



If we use (5.5.43) and the well-known trigonometric formulas for 
«/i/ 2 (^)> J -112(9) we see that the last equation is equivalent to 



dl 

dg 



~\l 2 — x 

+ (2i cos x)I = — 1/ — ^4 e~ iQ cos -. 

* 7T(0 2 



A solution i^ of the non-homogeneous equation which in general 
vanishes as q -> 00 is readily found: 



C 



£4 

gig cos 

V27T 



* COS - 

21 



00 „— iX{l+ cos x) 



dh 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 125 

Thus for / the appropriate solution of the differential equation 
must be 

j = e i Q cosx + / ^ 

Introduction of a new variable of integration z in the expression for 
I N through the relation 2X cos 2 x/2 = z 2 , and use of the formula 



f 



X 



COS 



e~ iz dz = \/tz e 4 
leads with no difficulty to the expression 

in 

eT c v&> 

(5.5.47) I(q, x) = e ie cos * 2 e~ iz * dz 

V 71 J-00 

and this leads, in conjunction with (5.5.45), to Sommerf eld's solution. 

To derive the asymptotic behavior of I(g, x) as g -> oo we proceed 

a little differently. The fractions in the square brackets in (5.5.46) 

are combined, and some algebraic manipulation is applied, to yield 

— (C 1/2 + i£~ 112 ) e~e\ V2 ) 
I(q, x) 



: f V 2 



1 *JL (fl/2 _ if-1/2) _ V2 /? COS - C 

A/2 2 



A new integration variable X is now introduced by the equation 

(C l/2 + iC -X/ 2) 



/M/2 .•j.-I/Sn ji "C / 



A 


= V2 


(£1/2 


- ^- 1/2 ) ? 


dX = 




with the 


result 










(5.5.48) 






/(£,*?) = 


e -lQ 

2ni % 


i; 



2V2 C 

X — a/2 e 4 cos - 
2 

The path P (cf. Fig. 5.5.6) is transformed into the path L shown in 
Fig. 5.5.7, as one readily can see. The path L leaves the circle of 
radius a/2 centered at the origin on its left. This representation of 
the function I(g, x) is obviously a good deal simpler than that 
furnished by (5.5.46), and it is quite advantageous in studying the 
properties of the solution: for one thing, the plane waves at oo 
can be obtained as the residues at the poles 

(5.5.49) X ± = V2 eT cos -±^. 



126 



WATER WAVES 



In fact, if there is a pole in the upper half of the A-plane (and there 
may or may not be, depending on the values of both 6 and a) one 



X-plane 




Fig. 5.5.7. The path L in the A-plane 

has, after deformation of the path L over it and into the real axis, 
for I(q, k) the result: 

e~ Qk * dl 
(5.5.50) I(q,x) = e-**#* eo " i:zi 






plQ COS X 



e 

2711 



iQ /» oo 



3m K 

X — a/2 e 4 cos - 
2 

-** dl 



a/2 e 4 cos 



If k = jt — the only case in which there is a singularity on the real 



z-plane 




Fig. 5.5.8. The path C in the z-plane 

axis, i.e. a pole at X = — we assume that the path of integration 
is deformed near the origin into the upper half-plane. It is convenient 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



127 



to introduce the variable z 
(5.5.51) I(g,x) = ^ cos * + 



gX 2 in the integral, with the result 

e -iQ f e -z fa 

4>ni 



,-iq r e ~z 

WqJcJ\ 



[0 



with X x = \/2 e iSn/4: cos x/2, and C the path of integration shown 
in Fig. 5.5.8 For large values of g, and assuming X x ^ 0, the square 
bracket in the integrand can be developed in powers of (z/g) 112 , and 
we mav write 



h 



e~ z dz 



© 



-k 






dz 



K\q) ~ J 



i r e- z T 1 iz\ 



? 1 z 



dz. 



It is clear that we may allow e -> (see Fig. 5.5.8) and hence the 
path C can be deformed into the two banks of the slit along the 
real axis; each of the terms in the square brackets then can be 
evaluated in terms of the .T-function (cf., for example, MacRobert 
[M.2], p. 143). It is thus clear that for X x ^ 0, the leading term in 
the asymptotic expansion of the integral in (5.5.51) behaves like 
l/\/g; in fact, we have for I(g, x): 

e -iQ _ vn 1 ) 

I(g,x)~e iQCOS * 



(5.5.52) 
Since /U 



4mA. 



Vq 



\/ji and A, 



iZn 



a/2 e 4 cos - we have 
2 



(5.5.53) 



I(g, x)~ e iQ 



-tQ — l- 

4 



2 A/2 



Tig cos 



Of course, this holds only if x lies in the range ^ x < n since a pole 
occurs in the upper half of the A-plane only when cos x/2 is positive 
(cf. (5.5.50)). We must also exclude the value x = n, corresponding 
to X x = 0. Since x = 6 T a, we see that the values = n ± a cor- 
respond to the exceptional value x = n, and these values of d, in 
turn, are those which yield the lines in the physical plane across 
which our solution / behaves discontinuously at oo. (Cf. Fig. 5.5.4). 



128 WATER WAVES 

The discussion of the last paragraph yields the result, in conjunction 
with equation (5.5.45) which defines our solution in terms of I(g, x): 

(5.5.54) f(g, 6) ~ e iQ cos (0 - a) + e iQ cos (0+a) 

n n 

e~ lQ - l l e- iQ - l l 



., /7T- 6 - (x n / d + a 

2V27ZQ COS 2V27ZQ COS — — 

— A 

for large q and for angles d such that < 6 < n — a, and a in the 
range < a < n: only in this case are there poles of both of the 
integrals in (5.5.45) in the upper halfplane. 

The discussion of the behavior of the solution in other sectors of 
the physical plane and along the exceptional lines can be carried out 
in the same way as above. For example, if X + = \/2 e i3nli cos (6 +a/2 ) = 0, 
and hence d = n — a, it follows that there is only one pole in the 
upper halfplane and our solution f(g, ti—ol) is given by (cf. 5.5.48), 
(5.5.49)): 

e -ig i* oo p-Q*- 2 

f(g, n - a) = e iQCO * (7I - 2a > -.zri] 

J —00 



2jii J_ 00 X — X_ 



e -iQ r e -QA- 

+ : -—dX 

2jziJ L X 

or also (cf. (5.5.51) and Fig. 5.5.8) by: 

p—iQ r p~ z rl7 

f(g,7T-*) = eie™^+ -1— \ 

Q ll2 Jc 



^rJci 



© 



e~ te i e 



+ —\—d*. 

4<7ZlJ c z 



The asymptotic behavior of / can now be determined in the same 
way as above; the result is 

in 

(5.5.55) f(g, 7i - a) ~ ^ cos {n ~ 2a) + 1 e 



2 n /- — 71 - 2a 
2 V 2tzq cos 

the second term resulting from the pole at the origin. 

In this fashion the behavior of f(g, 6) for large values of q is deter- 
mined, and leads to 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



129 



5.5.56) 

f(Q,6)~g(Q 9 6) 



e iq cos (9— a) _i_ e iQ cos (0+a) 
e iQ cos (n-2<x) I Ig—iQ 



iQ cos (0— a) 



Ip-iQ 



<d <Tl 



6 = n — a 

7T — a < < tt + a 

= te + a 

7r + a < 6 < 2n. 

This is, of course, a verification of one of the conditions imposed 
at oo. In addition, the next terms in the asymptotic expansion, of 
order ljy/Q, are also determined, as follows: 



0<Q<7i— a 



(5.5.56)' 

f(e,0)-g( Q ,6) 



2V271Q 



cos 



2V271Q 



COS 



6 + a 



2V / 2tt£ 



71 



2a 



COS 



/ 6 - a / (9 + a 

2V2tt£ cos 2V27T^ cos 



, 6=7Z — QL 



, 7l — 0L<6<7l-{-G(. 



2V271Q 



7i + 2a 



2V2nq cos 



/ 6 +a 

2V27TO cos 

* 2 



0=7r+a 



7r+a<0<27i 



We observe that these expansions do not hold uniformly in because 
of zeros in the denominators for 6 = n db a, i.e. at the lines of 
discontinuity of the function g(@, 0). 

With the aid of the function g(g, 6) defined in (5.5.56) we define 
a function h(g, d) by the equation 
(5-5.57) f(Q,0)=g( Qt 0) +h( Q> 6). 

Thus h is of necessity a discontinuous function since / is continuous 
while g has jump discontinuities along the lines = n ± a. The 
function h is given by (cf. (5.5.5.0)): 



130 



(5.5.58) h(g,d) 



WATER WAVES , 




- e~ ie r * e~^ dX 


f" °° e-e* 2 dX 


2m J_oo X - X_ J 


-*X-X + 



with the proviso that the integrals should be deformed into the 
upper half-plane in the vicinity of the origin in case either X_ or X + 
vanishes: i.e., in case 6 has one of its two critical values n ± a. 
That the sum g + h really is our solution / is rather clear in the 
light of our discussion above; and that it has jump discontinuities 
which just compensate those of g in order to make / continuous can 
also be easily verified. We shall not carry out the calculation here. 
The function h(g, 0), in view of (5.5.56) and (5.5.57) thus yields what 
might be called the "scattered" part of the wave. 

In order to show that our solution / satisfies the conditions of the 
uniqueness theorem proved above, we proceed to show that h as 
defined by (5.5.58) satisfies the radiation condition (5.5.9); afterwards 
we will prove that / behaves at the origin as prescribed by (5.5.11). 
Our solution / will thus be proved to be unique. 

That the function h(g, 6) defined by (5.5.58) satisfies the radiation 
condition is not at all obvious: one sees, for example (cf. (5.5.56)'), 
that its behavior at go is far from being uniform in the angle 6. 
In fact, the transformation of h to be introduced below is motivated 
by the desire to obtain an estimate for the quantity | dhjdg + ih \, 
which figures in the radiation condition, that is independent of d; 
and this in turn means an estimate independent of the quantities 
X_ and X + defined by (5.5.49). The function h(g, 6) can first of all 
be put in the form 

as one readily verifies. We proceed as follows: First we write 



h(e,0) 



Til 



/»Q0 /»« 

Jo Jo 



e-^-^dtdX +X 



l*oo (*a 

Jo Jo 



W-WdtdX 



then carry out the integrations with respect to X to obtain 



h(Q,6)=- 

2Vni 

From this representation of 

dh er« 

— + ih = — 

OQ Wjii 



■*dt 



h we obtain 



Jc 



A* dt 



(o + t)2J 



r e^dt T 00 e x+t dt " 

Jo (Q +t)\ Jo (o +t)i- 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



131 



It is important to observe that X 2 _ and Ai have pure imaginary values, 
as we know from (5.5.49). We also observe that the exceptional 
lines 6 = n ± a, which correspond to X T =0, simply have the effect 
that one of the two terms in the brackets in the last equation vanishes. 
From the Schwarz inequality we have 



I dh , ■* 



:C 



8tt 



J" 

Jo 



,A t 



dt 
(Q+t)l 



+ 



Consider the first term on the right; 



we find: 



f 

Jo 



xt *dt 



e A + 



(e+0i 



i 



!*-* d* 



2 



^IA- 



o (p+W 



Jo 



<ft 



< 



2 



3 



-If 



d< 



o (e + 0« 



/ 2 °°\ /l 8 r°° eft \ 

\ (p +01 0/ Vol 2 Jo (n +t)\) 



(Q+*. 



< 



Since the same estimate holds for the second term, it follows that 



dh 

dQ 



+ ih 



< 



1 

71Q' 



and this estimate holds for all values of d, since it holds for the two 
exceptional values 6 = n i a as well as for all other values in the 
range 5^ 5^ 2n. We have thus verified that the radiation condition 
holds — in fact, we have shown that it holds in the strong form. 
We proceed to show that /(p, 6) behaves properly at the origin. 
To this end, we start with the solution in the form (cf. (5.5.48) and 
(5.5.45)): 

e-^\C e-fdl r e~fdX\ 



with L the path of Fig. (5.5.7). The transformation X = -y/z is then 
made, so that the new path of integration D is like the path C in 
Fig. 5.5.8 except that the circular part now has a radius large enough 
to include the singularities of the integrands in its interior. We may 
take the radius of the circular part of D to have the value 1/p, since 
we care only for small values of p in the present consideration. The 
transformation gz = u then leads to the following formula for /(p, 6): 

n ) _ e ~ i9 j( e~ u du r e~ u du 

4>7ii \Jd 1 u H u ^ — A-gt) Jd u?(u? — X + gi 



132 



WATER WAVES 



with D x a path of the same type as D except that the circular part 
of D 1 is now the circle of unit radius. For small values of g the integrals 
in the last expression can be expressed in the form 






1 + 



Q™k ± , Q)? ± 



uh 



+ 



2ni 



1 + Q*l ± j 



du 



du -f- 



112 






du 4- . 



From this expansion we see clearly that 

7(e,e)~i 



/,(e.O) — i 



Q ->0. 



This completes the verification of the conditions needed for the 
application of the uniqueness theorem to our solution /. 

It has been shown by Putnam and Arthur [P. 18] (see also Carr 
and Stelzriede [C.l]) that the theory of diffraction of water waves 




Fig. 5.5.9. Waves behind a breakwater 

around a vertical barrier is in good accord with the physical facts, 
the accuracy being particularly high in the shadow created by the 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 133 

breakwater. Figure 5.5.9 is a photograph (given to the author by 
J. H. Carr of the Hydrodynamics Laboratory at the California 
Institute of Technology) of a model of a breakwater which gives some 
indication of the wave pattern which results. 



5.6. Brief discussions of additional applications and of other methods 
of solution 

The object of the present section is to point out a few further 
problems and methods of dealing with problems concerned, for the 
most part, with simple harmonic waves of small amplitude. 

The first group of problems to be mentioned belongs, generally 
speaking, to the field of oceanography. For general treatments of 
this subject the book of Sverdrup, Johnson, and Fleming [S.32] 
should be consulted. One type of problem of this category which 
was investigated vigorously during World War II is the problem of 
wave refraction along a coast, or, in other terms, the problem of 
the modification in the shape of the wave crests and in the amplitude 
of ocean waves as they move from deep water into shallow water. 
We have seen in the preceding sections that it is not entirely easy 
to give exact solutions in terms of the theory of waves of small 
amplitude even in relatively simple cases, such, for example, as the 
case of a uniformly sloping bottom. As a consequence, approximate 
methods modeled after those of geometrical optics were devised, 
beginning with the work of Sverdrup and Munk [S.35]. Basically, 
these methods boil down to the assumption that the local propagation 
speed of a wave of given length is known at any point from the for- 
mulas derived in Chapter 4 for water of constant depth once the 
depth of the water at that point is known; and that Huygens' principle, 
or variants of it, can be used to locate wave fronts or to construct 
the rays orthogonal to them. The errors resulting from such an as- 
sumption should not be very great in practice since the depth varia- 
tions are usually rather gradual. Various schemes of a graphical 
character have been devised to exploit this idea, for example by 
Johnson, O'Brien, and Isaacs [J. 7], Arthur [A.3], Munk and Traylor 
[M.16], Suquet [S.30], and Pierson [P.8]. Figure 5.6.1 is a refraction 
diagram for waves passing over a shoal in an otherwise level bottom 
in the form of a flat circular hump, and Fig. 5.6.2 is a picture of the 
actual waves. Both figures were taken from a paper by Pierson [P.8], 
and they refer to waves in an experimental tank. As one sees, there 



134 



WATER WAVES 




Fig. 5.6.1. Theoretical wave crest-orthogonal pattern for waves passing over a 
clock glass. No phase shift 



is fair general agreement in the wave patterns— even good agreement 
in detail over a good part of the area. However, near the center of 
the figures there are considerable discrepancies, since the theoretical 
diagram shows, for instance, a sharp point in one of the wave crests 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



135 



which is lacking in the photograph. The fact is that there is a caustic 
in the rays constructed by geometrical optics (i.e. the orthogonals 
to the wave crests have an envelope), and in the vicinity of such a 
region the approximation by geometrical optics is not good. One of 



* t 



2 



» 



F 




• ft ■ 


f < # 






7 




/ / 




{ * m 


% it 


v* 


*. ** '■'■'■ 


V 


*w 



?'-' 



! i i * 

in] 



Fig. 5.6.2. Shadowgraph for waves of moderate length passing over a clock glass 

the interesting features of Fig. 5.6.2 is that the shoal in the bottom 
results in wave crests which cross each other on the lee side of the 
shoal, although the oncoming waves form a single train of plane 
waves. Figure 5.6.3 is an aerial photograph (again taken from the 



I 



136 



WATER WAVES 



paper by Pierson) showing the same effect in the ocean at a point 
off the coast of New Jersey; the arrow points to a region where there 
would appear to be three wave trains intersecting, but all of them 
appear to arise from a single train coming in from deep water. 




Fig. 5.6.3. Aerial photograph at Great Egg Inlet, New Jersey 

In the case of sufficiently shallow water Lowell [L.16] has studied 
the conditions under which the approximation by geometrical optics 
is valid; his starting point is the linear shallow water theory (for 
which see Ch. 10.13) in which the propagation speed of waves is 
Vgh, with h the depth of the water, and it is thus independent of 
the wave length. Eckart [E.2, 3] has devised an approximate theory 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 137 

which makes it possible to deal with waves in both deep and shallow 
water, as well as in the transition region between the two. 

There is an interesting application of the theory of water waves to 
a problem in seismology which will be explained here even though it 
is necessary to go somewhat beyond the linear theory on which this 
part of the book is based. We have seen in Chapter 3 above that the 
displacements, velocities, and pressure variations in a simple harmonic 
standing wave die out exponentially in water of infinite depth. 
However, it was pointed out by Miche [M. 8] that this is true only of 
the first order terms in the development of the basic nonlinear theory 
with respect to the wave amplitude; if the development is carried out 
formally to second order it turns out that the pressure fluctuates 
with an amplitude that does not die out with the depth, but depends 
on the square of the amplitude. (For progressing waves, this is not 
true. ) In addition, the second order pressure variation has a frequency 
which is double the frequency of the linear standing wave. (It is not 
hard to see in a general way how this latter nonlinear effect arises 
mathematically. In the Bernoulli law, the nonlinear term of the form 
0\ + 0\ would lead, through an iteration process starting with 
= Ae my cos mx cos ot, to terms involving cos 2 at and thus to 
harmonics with the double frequency. ) It happens that seismic waves 
in the earth of very small amplitudes — called microseisms — and of 
periods of from 3 to 10 seconds are observed by sensitive seismo- 
graphs; these waves seem unlikely to be the result of earthquakes or 
local causes; rather, a close connection between microseisms and dis- 
turbed weather conditions over the ocean was noticed. However, since 
it was thought that surface waves in the ocean lead to pressure 
variations which die out so rapidly in the depth that they could not 
be expected to generate observable waves in the earth, it was thought 
unlikely that storms at sea could be a cause for microseisms. The result 
of Miche stated above was invoked by Longuet-Higgins and Ursell 
[L. 14] in 1948 to revive the idea that storms at sea can be the origin 
of microseisms. (See also the paper of 1950 by Longuet-Higgins 
[L.13].) In addition, Bernard [B.8] had collected evidence in 1941 
indicating that the frequency of microseisms near Casablanca was 
just double that of sea waves reaching the coast nearby; the same 
ratio of frequencies was noticed by Deacon [D. 6] with respect to 
microseisms recorded at Kew and waves recorded on the north coast 
of Cornwall. Further confirmation of the correlation between sea 
waves and microseisms is given in the paper of Darbyshire [D. 4]. 



138 WATER WAVES . 

A reasonable explanation for the origin of microseisms thus seems to be 
available. Of course, this explanation presupposes that standing 
waves are generated, but Longuet-Higgins has shown that the needed 
effects are present any time that two trains of progressing waves 
moving in opposite directions are superimposed, and it is not hard to 
imagine that such things would occur in a storm area— for example, 
through the superposition of waves generated in different portions of 
a given storm area. It might be added that Cooper and Longuet- 
Higgins [C.3] have carried out experiments which confirm quantita- 
tively the validity of the Miche theory of nonlinear standing waves. It 
is perhaps also of interest to refer to a paper by Danel [D.2] in which 
standing waves of large amplitude with sharp crests are discussed. 

In Chapter 6 some references will be made to interesting studies 
concerning the location of storms at sea as determined by observa- 
tions on shore of the long waves which travel at relatively high speeds 
outward from the storm area (cf. the paper by Deacon [D.6]). 

The general problem of predicting the character of wave conditions 
along a given shore is, of course, interesting for a variety of reasons, 
including military reasons (see Bates [B.6], for example). Methods 
for the forecasting of waves and swell, and of breakers and surf are 
treated in two pamphlets [U.l, 2] issued by the U.S. Hydrographic 
Office. 

A necessary preliminary to forecasting studies, in general, is an 
investigation of ways and means of recording, analyzing, and repre- 
senting mathematically the surface of the ocean as it actually occurs 
in nature. Among those who have studied such questions we mention 
here Seiwell [S.9, 10] and Pierson [P.10]. The latter author concerns 
himself particularly with the problem of obtaining mathematical re- 
presentations of the sea surface which are on the one hand sufficiently 
accurate, and on the other hand not so complicated as to be practically 
unusable. The surface of the open sea is, in fact, usually extraordinar- 
ily complicated. Figure 5.0.4 is a photograph of the sea (taken from 
the paper by Pierson) which bears this out. Pierson first tries re- 
presentations employing the Fourier integral and comes to the con- 
clusion that such representations would be so awkward as to preclude 
their use. (In Chapter 6 we shall have an opportunity to see that it is 
indeed not easy to discuss the results of such representations even for 
motions generated in the simplest conceivable fashion — by applying 
an impulse at a point of the surface when the water is initially at rest, 
for example. ) Pierson then goes on to advocate a statistical approach 



WAVES OX SLOPING BEACHES AND PAST OBSTACLES 



139 



to the problem in which various of the important parameters are 
assumed to be distributed according to a Gaussian law. These deve- 
lopments are far too extensive for inclusion in this book — besides, the 




Fig. 5.6.4. Surface waves on the open sea 

author is, by temperament, more interested in deterministic theories 
in mechanics than in those employing arguments from probability 
and statistics, while knowing at the same time that such methods are 
very often the best and most appropriate for dealing with the com- 
plex problems which arise concretely in practice. It would, however, 
seem to the author to be likely that any mathematical representations 
of the surface of the sea — whether by the Fourier integral or any other 
integrals — would of necessity be complex and cumbersome in propor- 
tion to the complexity of that surface and the degree to which details 
are desired. 

Before leaving this subject, it is of interest to examine another pho- 
tograph of waves given by Pierson [P. 10], and shown in Fig. 5.6.5. 
Near the right hand edge of the picture the wave crests of the pre- 
dominant system are turned at about 45° to the coast line, and they 
are broken rather than continuous; such wave systems are said to be 
short-crested. About half-way toward shore it is seen that these 
waves have arranged themselves more nearly parallel to the coast 
(indicating, of course, that the water has become shallower) and at 
the same time the crests are longer and less broken in appearance, 



140 



WATER WAVES 




Fig. 5.6.5. Aerial photograph over Oracoke 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 141 

though no single one of them can be identified for any great distance. 
Near the shore, the wave crests are relatively long and nearly parallel 
to it. On the photograph a second train of waves having a shorter 
wave length and smaller amplitude can be detected; these waves are 
traveling almost at right angles to the shore (they are probably caused 
by a breeze blowing along the shore) and they are practically not 
diffracted. Each of the two wave trains appears to move as though 
the other were not present: the case of a linear superposition would 
thus seem to be realized here. One observes also that there is a shoal, 
as evidenced by the crossed wave trains and the white-water due to 
breaking over the shoals. 

We pass next to a brief discussion of a few problems in which our 
emphasis is on the methods of solution, which are different from those 
employed in the preceding chapters of Part II. The first such problem 
to be discussed employs what is called the Wiener-Hopf method of 
solving certain types of boundary problems by means of an ingenious, 
though somewhat complicated, procedure which utilizes an integral 
equation of a special form. This method has been used, as was men- 
tioned in the introduction to this chapter, by Heins [H.12, 13] and by 
Keller and Weitz [K.9] to solve the dock problem and other problems 
having a similar character with respect to the geometry of the domains 
in which the solution is sought. However, it is simpler to explain the 
underlying ideas of the method by treating a different problem, i.e. 
the problem of diffraction of waves around a vertical half-plane — in 
other words, Sommerfeld's diffraction problem, which was treated by 
a different method in the preceding section. We outline the method, 
following the presentation of Karp [K.3]. The mathematical formula- 
tion of the problem is as follows. A solution <p(x, y) of the reduced 
wave equation 

(5.6.1) S7 2 <p+k 2 (p = 

is to be found subject to the boundary condition 

(5.6.2) (p y = for y = 0, x > 

and regular in the domain excluding this ray (cf. Fig. 5.6.6). In addi- 
tion, a solution in the form 

(5.6.3) cp = cp -f 9?j 
with (p defined by 

(5.6.4) (p Q = £.**(* cos O + V sin e ) } <d < 2jz, 



142 



WATER WAVES 



and with 9^ prescribed to die out at 00 is wanted. In other words, a 
plane wave comes from infinity in a direction determined by the angle 
Q , and the scattered wave caused by the presence of the screen, and 




Fig. 5.6.6. Diffraction around a screen 



given by (p x , is to be found. It is a peculiarity of the Wiener-Hopf 
method — not only in the present problem but in other applications to 
diffraction problems as well — that the constant k is assumed to be a 
complex number (rather than a real number, as in the preceding 
section) given, say, by k = k x + ih 2 , with k 2 small and positive. With 
this stipulation it is possible to dispense with conditions on q> x of the 
radiation type at 00, and to replace them by boundedness conditions. 
We employ a Green's function in order to obtain a representation 
of the solution in the form of an integral equation of the type to which 
the Wiener-Hopf technique applies. In the present case the Green's 
function G(x, y; x , y ) is defined as that solution of (5.6.1) in the 
whole plane which has a logarithmic singularity at the point (cc , y ) 
and dies out at 00 (here the fact that k is complex plays a role). This 
function is well-known; it is, in fact, the Hankel function H^(kr) 
of the first kind: 



(5.6.5) G(x, y;x ,y ) 



HfPikUx-XtF + to-y,)*]*). 



The next step is to apply Green's formula to the functions 99 and G in 
the domain bounded by the circle C 2 and the curves marked C t in 
Fig. 5.6.6. Because of the fact that G is symmetric, has a logarithmic 
singularity, and that 9? and G both satisfy (5.6.1), it follows by argu- 
ments that proceed exactly as in potential theory in similar cases that 
cp (x, y) can be represented in the form 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 



143 



(5.6.6) cp{x, y) 



f 

Jo 



w 



dG 



dx + exp {ik(x cos 6 + ?/ sin O )} 



when the radius of the circle C 2 is allowed to tend to oo, and the boun- 
dary condition (5.6.2), the regularity conditions, and conditions at oo 
are used. (A mild singularity at the edge x = 0, y = of the screen 
must also be permitted.) The symbol [<p] under the integral sign re- 
presents the jump in cp across the screen, which is of course not known 
in advance. The object of the Wiener-Hopf technique is to determine 
[99] by using the integral equation (5.6.6); once this is done (5.6.6) 
yields the solution (p(x, y). The first step in this direction is to differen- 
tiate both sides of (5.6.6) with respect to y, then set y = and 
confine attention to positive values of x; in view of the boundary 
condition (5.6.2) we obtain in this way the integral equation 



(5.6.7) ( 
The kernel K(x 
(5.6.8) 



ik sin 6 e 



ikx cos 



+ 



/•oo 

[cp(x )]K(x - x )dx , 
Jo 



x > 0. 



of the integral equation 
d 2 G 
dydyo y = i/ = o 



given by 



K(x — x ] 



Equation (5.6.7) is a typical example of an integral equation solvable 
by the Wiener-Hopf technique; its earmarks are that the kernel is a 
function of (x — x ) and the range of integration is the positive real 
axis. 

The starting point of the method is the observation that the integral 
in (5.6.7) is strongly reminiscent of the convolution type of integral 
in the theory of the Fourier transform. In fact, if the limits of integra- 
tion in (5.6.7) were from — 00 to + 00 and the equation were valid 
for all values of x, it could be solved at once by making use of the 
convolution theorem. This theorem states that if 

— fee , f*00 

/(a) = f(x ) exp{— ioLX Q }dx and K(<x.)= \ K(x ) exp {— i<xx }dx 

J —00 J —00 

— i.e. if/ and K are the Fourier transforms of/ and k (cf. Chapter 6) — 



_ /•OO 

then f(<x.)K((x.) -— f(x )K(x — x )dx , in other words, the transform 

J — OC 

of the integral on the right is the product of the Fourier transforms 
of the function f(x ) and K(x ) (cf. Sneddon [S.ll], p. 24). Conse- 
quently if (5.6.7) held in the wider domain indicated, it could be 
used to vield 



144 WATER WAVES 



= h(x) + [<p(*)W(*), 
with h(a.) the transform of the nonhomogeneous term in the integral 
equation. This relation in turn defines the transform [99(a)] of 
[(p(cc )] since ^(a) and K(ol) are the transforms of known functions, 
and hence [(p(x )] itself. We are, of course, not in a position to proceed 
at once in this fashion; but the idea of the Wiener-Hopf method is to 
extend the definitions of the functions involved in such a way that 
one can do so. To this end the following definitions are made 

g(x) = 0, x>0; f(x ) = [99], x > 

(5.6.9) • h(x) = 0, x < 0; f(x ) = 0, x < 

h(x) = ik sin d e ikx cos \ x > 0. 

Equation (5.6.7) can now be replaced by the equivalent equation 

(5.6.10) g(x) = h(x) + f(x )K(x — x )dx , — 00 < x < 00. 

J-00 

Here g(x) is unknown for x < and f{x ) — the function we seek — is, 
of course, unknown for x > 0; thus we have only one equation for 
two unknown functions. Nevertheless, both functions can be 
determined by making use of complex variable methods applied to 
the Fourier transform of (5.6.10); we proceed to outline the method. 
We have, to begin with, from (5.6.10): 

(5.6.11) g(a). = S(a) + /(a)Z(a), 
with ^(a) and K(ol) known functions given by 

-, ^ k sin d a 

(5.6.12) h(*)= °—, 

a — k cos u 

(5.6.13) K(ol) = —(k 2 - a 2 )i 

The equation (5.6.11) is next shown to be valid in a strip of the com- 
plex a-plane which contains the real axis in its interior. We omit the 
details of the discussion required to establish this fact; it follows in an 
elementary way from the assumption that the constant k has a posi- 
tive imaginary part, and from the conditions of regularity and boun- 
dedness imposed on the solution 99 of the basic problem. K(ol) is 
factored* in the form (i/2)K_((x.) -K + {ol) with K_(ol) = (k - a) 1/2 , 
K+(ol) = (k + a) 1/2 with K_ and K + regular in lower and upper half- 

* Such a manipulation occurs in general in using this technique; usually a 
continued product expansion of the transform of the kernel is required. 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 145 

planes, respectively. The equation (5.6.11) can then be expressed, 
after some manipulation, in the form 

(5 6 14) g+< a ) _ k sin 6 o [ 1 _ L " 

(*■ + a) 1/2 a - k cos O [_(k + a) 1/2 (* + &cos d ) ll2 _ 

ksmd i - , 

= ; — + — f-M(k — a) 1/2 

(k + k cos o ) 1/2 (a - k cos O ) 2 ' 

where the symbols g + and /_ refer to the fact that g(a) and /(a) can be 
shown to be regular in upper and lower half-planes of the complex 
a-plane, respectively, each of which overlaps the real axis. In fact, the 
entire left side of (5.6.14) is regular in such an upper half-plane, and 
similarly for the right hand side in a lower half-plane. Thus the two 
sides of the equation define a function which is regular in the entire 
plane, or, in other words, each side of the equation furnishes the 
analytic continuation of the function defined by the other side. 
Finally, it is rather easy to show, by studying the behavior of g(a) 
and /(a) at oo, that the entire function thus defined tends uniformly 
to zero at oo; it is therefore identically zero. Thus (5.6.14) defines both 
g(a) and /(a) since they can be obtained by equating both sides se- 
parately to zero. Thus g(x) and f(x) are determined, and the problem 
is, in principle, solved. 

The Wiener-Hopf method is, evidently, a most amusing and in- 
genious procedure. However, it also has somewhat the air of a tour de 
force which uses a good many tools from function theory (while 
the problem itself can be solved very nicely without going into the 
complex domain at all, as we have seen in the preceding section) and 
it also employs the artificial device of assuming a positive imaginary 
part for the wave number k. (This brings with it, we observe from 
(5.6.4), that while the primary wave dies out as x -» + oo, it be- 
comes exponentially infinite as x -» — oo.) In addition, the problem 
of diffraction by a wedge, rather than by a plane barrier, can not be 
solved by the Wiener-Hopf method, but yields readily to solution by 
the simple method presented in the preceding section. The author 
hazards the opinion that problems solvable by the Wiener-Hopf 
technique will in general prove to be solvable more easily by other 
methods— for example, by more direct applications of complex in- 
tegral representations, perhaps along the lines used to solve the 
difficult mixed boundary problem treated in section 5.4 above. 

We mention next two other papers in which integral equations are 



146 WATER WAVES 

employed to solve interesting water wave problems. The first of these 
is the paper by Kreisel [K.19] in which two-dimensional simple 
harmonic progressing waves in a channel of finite depth containing 
rigid reflecting obstacles are treated. Integral equations are obtained 
by using an appropriate Green's function; Kreisel then shows that 
they can be solved by an iteration method provided that the domain 
occupied by the water does not differ too much from an infinite strip 
with parallel sides. (Roseau [R.9] has solved similar problems for 
certain domains which are not restricted in this way. ) It is remarkable 
that Kreisel is able to obtain in some important cases good and useable 
upper and lower bounds for the reflection and transmission coeffi- 
cients. References have already been made to the papers by John 
[J. 5] on the motion of floating bodies. In the second of these papers 
the problem of the creation of waves by a prescribed simple harmonic 
motion of a floating body is formulated as an integral equation. This 
integral equation does not fall immediately into the category of those 
which can be treated by the Fredholm theory; in fact, its theory has 
a number of interesting and unusual features since it turns out that 
the homogeneous integral equation has non-trivial solutions which, 
however, are of such a nature that the nonhomogeneous problem 
nevertheless always possesses solutions. 

Various problems concerning the effect of obstacles on waves, and 
of the wave motions created by immersed oscillating bodies, have 
been treated in a series of notable papers by Ursell [U.3, 4, 5 and 
U.8, 9, 10]. Ursell usually employs the method of expansions in terms 
of orthogonal functions, or representations by integrals of the Fourier 
type, as tools for the solution of the problems. 

Finally, it should be mentioned that the approximate variational 
methods devised by Schwinger [S.5] to treat difficult problems in the 
theory of electromagnetic waves can also be used to treat problems in 
water waves (cf. Keller [K.7]). A notable feature of Schwinger's 
method is that it is a technique which concentrates attention on the 
quantities which are often of the greatest practical importance, i.e. 
the reflection and transmission coefficients, and determines them, 
moreover, without solving the entire problem. Rubin [R.13] has for- 
mulated the problem of the finite dock — which has so far defied all 
efforts to obtain an explicit integral representation for its solution — 
as a variational problem of a somewhat unconventional type, and 
proved the existence, on the basis of this formulation, of solutions be- 
having at oo like progressing waves. 



WAVES ON SLOPING BEACHES AND PAST OBSTACLES 147 

An interesting type of problem which might well have been dis- 
cussed at length in this book is the problem of internal waves. This 
refers to the occurrence of gravity waves at an interface between two 
liquids of different density. Such problems are discussed in Lamb 
[L.3], p. 370. The case of internal waves in media with a continuous 
variation in density has considerable importance also for tidal motions 
in both the atmosphere (cf. Wilkes [W.2]) and the oceans (cf. 
Fjeldstad [FA]). 



SUBDIVISION B 

MOTIONS STARTING FROM REST. TRANSIENTS. 

CHAPTER 6 
Unsteady Motions 

6.1. General formulation of the problem of unsteady motions 

In the region occupied by the water we seek, as usual, a harmonic 
function 0(x, y, z; t) which satisfies appropriate boundary conditions 
and, in addition, appropriate conditions prescribed at the initial in- 
stant t = 0. At the free surface we have the boundary conditions 

(6.1.1) -®y+rjt = \ 

1 for y = 0, t > 

(6.1.2) t+gr} =—p I 

Q ' 

in terms of the vertical elevation rj(x, z; t) of the free surface and the 
pressure p(x, z; t) prescribed on the surface. As always in mechanics, 
a specific motion is determined only when initial conditions at the 
time t = are given which furnish the position and velocity of all 
particles in the system. This would mean prescribing appropriate 
conditions on throughout the fluid at the time t = 0, but since we 
shall assume to be a harmonic function at t = as well as for t > 
it is fairly clear that conditions prescribed at the boundaries of the 
fluid only will suffice since is then determined uniquely throughout 
its domain of definition in terms of appropriate boundary conditions.* 
As initial conditions at the free surface, for example, we might there- 
fore take 

(6.1.3) rj(x,z;0) = h(x, z) \ 

} at y = 0, 

(6.1.4) rjt(x,z;0) =/ 8 0b,*) J 

with f 1 and / 2 arbitrary functions characterizing the initial elevation 
and vertical velocity of the free surface. 

In water wave problems it is of particular interest to consider cases 

* We shall see later on (sections 6.2 and 6.9) that the solutions are indeed 
uniquely determined when the initial conditions are prescribed only for the 
particles at the boundary of the fluid. 

149 



150 WATER WAVES 

in which the motion of the water is generated by applying an impul- 
sive pressure to the surface when the water is initially at rest. To 
obtain the condition appropriate for an initial impulse we start from 
(6.1.2) and integrate it over the small time interval 5^ t ^ r. The 
result is 

(6.1.5) pdt= — q@(cc, 0, z; r) — Qg\ r]dt, 

Jo "Jo 

since 0(x, y, z; 0) can be assumed to vanish. One now imagines that 
t -> + while p -> oo in such a way that the integral on the left tends 
to a finite value— the impulse / per unit area. Since it is natural to 
assume that r\ is finite it follows that the integral on the right vanishes 
as t -> + , and we have the formula 

(6.1.6) / = — q0(x, 0, z; 0+) 

for the initial impulse per unit area at the free surface in terms of the 
value of there. If / is prescribed on the free surface (together with 
appropriate conditions at other boundaries), it follows that 
0(x, y, z; 0+) can be determined, or, in other words, the initial velocity 
of all particles is known. 

It is also useful to formulate the initial condition on at the free 
surface appropriate to the case in which the water is initially at rest 
under zero pressure, but has an initial elevation rj(x, z; 0). The condi- 
tion is obtained at once from (6.1.2); it is 

(6.1.7) t (x, 0, z; + ) = - gri(x, z; + ), 

since p = for t > 0. Prescribing the initial position and velocity of 
the free surface is thus equivalent to prescribing the initial values of 
and its first time derivative t . From now on the notation + will 
not be used in formulating initial conditions — instead we shall simply 
write instead of + . 

6.2. Uniqueness of the unsteady motions in bounded domains 

It is of some interest to consider the uniqueness of the unsteady 
motions, for one thing because of the unusual feature pointed out 
in the preceding section: it is sufficient to prescribe the initial position 
and velocity, not of all particles, but only of those on the boundary. 
A uniqueness proof based on the law of conservation of energy will 
be given. 

To this end, consider the motion of a bounded volume of water con- 
fined to a vessel with fixed sides but having a free surface (cf. Fig. 



UNSTEADY MOTIONS 



151 



6.2.1). In Chapter 1 we have already discussed the notion of energy 
and its time rate of change with the following results. For the energy 
E itself we have, obviously: 




y=h(x,z) 



Fig. 6.2.1. Water contained in a vessel 
(6.2.1) E(t) = q jjj [i(0 2 x +0 2 y + 2 z ) + gy] dx dy dz. 

R 

Here R refers to the volume occupied by the water at any instant. 
The x, z-plane is, as usual, taken in the plane of equilibrium of the 
free surface, and y = r\{x, z; t) and y = h(x, z) are assumed to be the 
equations of the free surface and of the containing vessel, respectively. 
The expression for E can now be written in the form 



(6.2.2) E(t) 



(J) 



(0 2 x + ®l + $>\) dx dy dz 



g \j(rj 2 - h 2 ) dx dz 



By S is meant the projection on the x, z-plane of the free surface and 
the containing vessel. In Chapter 1 the following expression for the 
rate of change of the energy E was derived: 

dE 



(6.2.3) 



dt 



ft 



= \\[Q®t{®n ~V n ) ~pV n ] dS. 



152 WATER WAVES 

By R is meant the boundary surface of R, while v n means the normal 

component of the velocity of R. It is essential for our uniqueness 

proof to observe that in the special case in which p = on the free 

surface we have 

dE 
(6.2.4) — = 0, E = const. 

This follows at once from the fact that v n = @ n = on the fixed part 
of the boundary, while v n = n and p = on the free surface. 

So far no use has been made of the fact that we consider only a 
linear theory based on the assumption of small oscillations about the 
equilibrium position. Suppose now that the initial position and velo- 
city of the water particles has been prescribed, or, as we have seen in 
the preceding section that r\(x, z; 0) and 0(x, y, z; 0) are given func- 
tions: 

tj(x, z; 0) = f x (x, z) 



1 0(x, y, z; 0) = f 2 (x, y, z). 

We proceed next in the customary way that one uses to prove unique- 
ness theorems in linear problems. Suppose that rj v & l9 and r} 2 , 2 
are two solutions of the initial value problem. Then = 1 — 2 
and r\ = 7} 1 — f] 2 are functions which satisfy all of the conditions 
originally imposed on { and r\i except that f x and f 2 in (6.2.5) would 
now both vanish, and the free surface pressure would also vanish 
(cf. (6.1.2) and (6.1.7)). (Here the linearity of our problem is used in 
an essential way. ) It follows therefore that dEjdt = 0, and E = const, 
when applied to and rj, as we have seen. But at the initial instant 
7] = and = 0, so that 

(6.2.6) E = - — I h 2 dxdz, 

s 
from (6.2.2) as applied to<Z> = X — 2 and?y = q l — r\ 2 . Consequent- 
ly we have the result 

(6.2.7) j[[ (0l + 0\ + 0\) dx dydz+g [J V * dx dz = 0, 

and this sum obviously vanishes only if grad = and r\ = — in 
other words it follows that l = 2 (except for an unessential addi- 
tive constant), ?] 1 = rj 2 , and the uniqueness of the solution of the 
initial value problem is proved. 



UNSTEADY MOTIONS 153 

The proof given here applies only to a mass of water occupying a 
bounded region. Nevertheless, it seems clear that the uniqueness of 
the solution of the initial value problem is to be expected if the water 
fills an unbounded region, provided that appropriate assumptions 
concerning the behavior of the solution at oo are made. In the follow- 
ing, a variety of such cases will be treated by making use of the tech- 
nique of the Fourier transform and, although no explicit discussion 
of the uniqueness question will be carried out, it is well-known that 
uniqueness theorems (of a somewhat restricted character, it is true) 
hold in such cases provided only that appropriate conditions at oo 
are prescribed. Recently these uniqueness questions have been treated 
by Kotik [K.17] and Finkelstein [F. 3]. The latter, for example, proves 
the uniqueness of unsteady motions in unbounded domains in 
which rigid obstacles occur, and both writers obtain their uniqueness 
theorems by imposing relatively weak conditions at infinity. In sec. 9 
of this chapter the theory devised by Finkelstein will be discussed. 

6.3. Outline of the Fourier transform technique 

As indicated above, the solutions of a series of problems of unsteady 
motions in unbounded regions as determined through appropriate 
initial conditions will be carried out by using the method of the 
Fourier transform. The basis for the use of this method is the fact that 
special solutions of our free surface problems are given — in the 
case of two-dimensional motion in water of infinite depth, for example 

-by 

(6.3.1) 0(x, y; t) = e m y sin (at + a) cos m(x — r) 
with 

(6.3.2) a 2 = gm 

and for arbitrary values of a and t. From these solutions it is possible 
to build up others by superposition, for example, in the form 

(6.3.3) 0(x, y; t) = f°° h(a)e m y sin (at + <x)dm 

/»oo 

X /(t) cos m(x — t) dr, 

J — oo 

in which h(a) and /(t) are arbitrary functions. This in turn suggests 
that the Fourier integral theorem could be used in order to satisfy 
given initial conditions, since this theorem states that an arbitrary 



154 WATER WAVES 

function f(x) defined for — co < x < oo can be represented in the 
form 

(6.3.4) f(x) = - da. f(rj) cos tx(rj — x) dr\ 

71 J J -oo 

provided only that f(x) is sufficiently regular (for example, that f(x) 
is piecewise continuous with a piecewise continous derivative is more 
than sufficient) and that f(x) is absolutely integrable: 

(6.3.5) f°° \f(x) | dx < oo. 

J — 00 

Indeed, we see that if we set h(a) = 1/tz and a = :rc/2 in (6.3.3) we 
would have exactly the integral in (6.3.4) for t = and y = 0, and 
hence 0(«, 0; 0) would reduce to the arbitrarily given function f(x). 
Thus a solution would be obtained for an arbitrarily prescribed initial 
condition on 0. 

It would be perfectly possible to solve the problems treated below 
by a direct application of (6.3.4), and this is the course followed by 
Lamb [L.3] in his Chapter IX. Actually the problems were solved 
first by Cauchy and Poisson (in the early part of the nineteenth cen- 
tury), who derived solutions given by integral representations before 
the technique of the Fourier integral was known. It might be added 
that these problems were considered so difficult that they formed the 
subject of a prize problem of the Academie in Paris. 

We prefer, in treating these problems, to make use of the technique 
of the Fourier transform (following somewhat the presentation given 
by Sneddon [S.ll], Chapter 7) since the building up of the solution to 
fulfill the prescribed conditions then takes place quite automatically. 
However, the method is based entirely upon (6.3.4) and thus also 
requires for its validity that the functions f(x) to which the technique 
is applied should be representable by the Fourier integral. This is a 
restriction of a non-trivial character: for example, the basically im- 
portant solutions given by (6.3.1) are not representable by the Fou- 
rier integral. 

It is useful to express the Fourier integral in a form different from 
(6.3.4). We write 



1 

f(x) = - lim 

71 f-»oo./ 



f{r))drj cos s(r] — x)ds. 
Jo 



UNSTEADY MOTIONS 

I 



155 



But since £ cos s(r} — x)ds = 2 cos ^(^ — x)ds and 
\ sin 5(77 — x)ds = 0, we may write cos s(tj — x)ds - 
Y 2 exp {is(x — rj)} ds, and hence 

— e* saj <& f{rj)e~ ir]S drj. 

<?ij— 00 J— 00 



(6.3.6) 

We now set 

(6.3.7) 



/(*) 



/» 



1 r°° 

V27Tj_oo 



)tf-*' sa! rfiC 



and call /(s) the Fourier transform of /(#)• It follows at once from 
(6.3.6) that the original function f(x) is obtained from its transform 
J(s ) by the inversion formula 



(6.3.8) 



m 



1 r°° 



f(s) e isx ds. 



In our differential equation problems it will be essential to express 
the Fourier transform of the derivatives of a function in terms of the 
transform of the function itself. Consider for this purpose the trans- 
form of d n f/dx n and integrate by parts (which requires that d n f/dx n 
be continuous): 



\/27T 



*> Jn 



— 00 

1 



d n f 

dx n 



e -isx d x 



d"- 1 / 



dx n 



-\- is 



r°° d n -^f 

J_ ao dx n - 1 



dx 



If the (n — l)-st derivative is to possess a transform it must tend to 
zero at i 00 and hence we have 



(6.3.9; 



d n f 
dx n 



is — 

dx n ~ l 



that is, the transform of the n-th derivative is (is) times the transform 
of the (n — 1 )-st derivative. By repeated application of this formula 
we obtain the result 



156 WATER WAVES • 

(6.3.10) — - = («)"/ 

provided that f(x) and its first n derivatives are continuous and 
that all of these functions possess transforms. 

A rigorous justification of the transform technique used in the 
following for solving problems involving partial differential equations 
is not an entirely trivial affair (see, for example, Courant-Hilbert 
[C.10], vol. 2, p. 202 ff.). Such a justification could be given, but we 
shall not carry it out here. Indeed, it would be reasonable to take the 
attitude that one may proceed quite formally provided that one veri- 
fies a posteriori that the solutions obtained in this way really satisfy 
all conditions of the problem. This is usually not too difficult to do, 
and, since the relevant uniqueness theorems are available, this course 
is perfectly satisfactory. 

6.4. Motions due to disturbances originating at the surface 

We wish to determine first the motion in two dimensions due to the 
application of an impulse over a segment of the surface — a 5S x ^ a 
at t = when the water is at rest in the equilibrium position. We 
suppose the depth h of the water to be constant and that it extends to 
infinity in the horizontal direction. The velocity potential 0(x, y; t) 
must satisfy the following conditions. It must be a solution of the 
Laplace equation: 

(6.4.1) XX + yy = 0, — oo < x < oo, — h ^ y ^ 0, t ^ 0, 
satisfying the boundary conditions 

(6.4.2) tt +g0 y = O; y = 0, t>0 
and 

(6.4.3) y = 0, y = — h, t ^ 0. 

The first of these conditions states that the pressure on the free 
surface is zero for t > 0. As initial conditions we have, in view of 
(6.1.6), (6.1.7), and the assumed physical situation: 

1 

(6.4.4) 0(x, 0; 0) = I(x), 

Q 

(6.4.5) t (x, 0; 0) = 0, 

with I(x) the impulse per unit area applied to the free surface. In 



UNSTEADY MOTIONS 157 

addition, we must impose conditions at oo. These are that and 
its first two derivatives with respect to x, y, and t should tend to zero 
at oo in such a way that all of these functions possess Fourier trans- 
forms with respect to x. This, in particular, requires that I(x) in 
(6.4.4) should vanish at oo. Actually we consider only the special 
case in which 

_., . [ P = const., I x I < a 

(6.4.6) I(x) = 

{ * 10, | x- 1 > a, 

i.e. the case in which a uniform impulse is applied to the segment 
\x \ < a, the remainder of the surface being left undisturbed. 

The solution 0(x, y; t) will now be determined by applying the 
Fourier transform in x to the relations (6.4.1) — (6.4.5) with the object 
(as always in such problems) of obtaining a simpler problem for the 
transform 0(s, y; t) = cp(s, y; t). Once the transform cp has been found 
by solving the latter problem the inversion formula yields the solution 
0. We begin by applying the transform to (6.4.1), i.e. by multiplying 
by e~ isx and integrating over the interval — oo < x < oo; the result is 

(6.4.7) — s 2 <p{s, y; t) + cp yy (s, y; t) = 

in view of (6.3.10) and the assumed behavior of at oo. (Clearly, it is 
also necessary to suppose that the operation of differentiating 
twice with respect to y can be interchanged with the operation of 
integrating over the infinite interval.) This step already achieves 
one of the prime objects of the approach using a transform: the trans- 
form (p satisfies an ordinary differential equation instead of the partial 
differential equation satisfied by 0. The general solution of (6.4.7) is 

(6.4.8) (p(s, y, t) = A(s; t)e |s,y + B(s; t)e~^ 

in terms of the arbitrary "constants" A(s; t) and B(s; t). It is a simple 
matter to find the appropriate special solution that also satisfies the 
bottom condition (6.4.3), and from it to continue (just as is done in 
what follows ) in such a way as to find the solution for water of uniform 
depth. However, we prefer to take the case of infinite depth and to 
replace (6.4.3) by the condition that y -> when y -> — oo. The 
transform (p then also must have this property so that we obtain for 
cp(s, y; t) in this case the solutions 

(6.4.9) (p(s, y; t) = A(s; t)e ^ v . 



158 WATER WAVES . 

The transform is next applied to the free surface condition (6.4.2) to 
obtain 

(6.4.10) <p tt +g(f y = 0, y = 0, t > 

and upon insertion of (p(s, 0; t) from (6.4.9) we find for A(s; t) the 
differential equation 

(6.4.11) A tt +g\s\A = 0, t>0. 

Finally, the initial conditions must be taken into account. The trans- 
form of (6.4.5) leads, evidently, to the condition A t (s; 0) = 0, and 
the solution of (6.4.11) satisfying this condition is 

(6.4.12) A(s; t) = a(s) cos (Vg\s\ t) 

with a(s) still to be determined by using (6.4.4). From (6.4.4) we 
have <p(s, 0; 0) = — (1/q)I(s) in which/ (s) is, of course, the transform 
of I(x) as given by (6.4.4); hence a(s) = — (1/q)I(s) and we have for 
0(s, y; t) = (p(s, y; t) the result 

(6.4.13) 0(s, y; t) = - — I(s)e^ y cos (Vg\s\ t). 

9 

The inversion formula (6.3.8) then leads immediately to the solution 

1 



(6.4.14) 0(x,y;t) 



I (s)e^ v e isx cos (Vg\s\ t) ds. 



In our special case (cf. (6.4.6)) we have for I(s): 

P [ a 2P [ a 2Pa sin sa 

I (s) = — -= e~ tsx dx = cos sx dx = —= ' ' 

V27iJ -a V 2n Jo V 271 sa 

and hence finally for 0(x, y; t) the solution 

2Pa f°°sin sa w , /—.*■, 

(6.4.15) 0(x, y;t)= — e y cos sx cos (y/gs t)ds, 

Tig J sa 

as one can readily verify. For the free surface elevation we have 
(from (6.1.2)): 



1 - 2Pa 
(6.4.16) rj(x; t) = - - t = = lim 

g 71Q Vg »->0 J 



e sy cos sx 

sa 



sin (Vgs t)Vs ds. 



UNSTEADY MOTIONS 159 

One observes that the integrals converge well for all y < because of 
the exponential factor e sy . i.e. everywhere except possibly on the free 
surface. These formulas can now be used to obtain the solution for the 
case of an impulse concentrated on the surface at x = 0; one need only 
suppose that a -» while P -> oo in such a way that the total im- 
pulse 2Pa tends to a finite limit. For a unit total impulse we would 
then obviously obtain for and r\ the formulas: 

1 f 00 .— 

(6.4.17) 0(x, y;t)= — — e sy cos sx cos ( Vgs t) ds, 

KQ Jo 

(6.4.18) rj(x; t) = — — lim e sy cos sx sin ( Vgs t)\/s ds. 

neVgv^oJo 

(We define r\{x\ t) as a limit for y -> since the integral obviously 
diverges for y = 0. This would, however, not be necessary in (6.4.16).) 
By operating in the same way, one can easily obtain the solutions 
corresponding to the case of an initial elevation of the free surface at 
time t = 0, with no impulse applied. The only difference would be that 
in (6.4.4) would be assumed to vanish while t in (6.4.5) would be 
different from zero. We simply give the result of such a calculation, 
but only for the limit case in which the initial elevation is concen- 
trated at the origin. For and r\ the formulas are: 

v§ r . . ,./-., ^ 



v< 



(6.4.19) 0(x,y;t)=~- — e sy cos sx sin ( Vgs t) 

n Jo 

1 f 00 /— 

(6.4.20) rj(x; t) = - lim e sy cos sx cos ( Vgs t) ds. 

71 v^oJo 

There is no difficulty in treating problems having cylindrical sym- 
metry that are exactly analogous to the above two-dimensional cases. 
In these cases also one could begin with the solutions having symmetry 
of this type that are simple harmonic in the time (cf. Chapter 3): 

(6.4.21 ) 0(r, y; t) = e iat e my J (mr) 

with o 2 = gm (for water of infinite depth). Here the quantity r is the 
distance Vx 2 -f z 2 from the y-axis, and J (mr) is the Bessel function 
of order zero that is regular at the origin. One could now build up more 
complicated solutions by superposition of these solutions and satisfy 
given initial conditions by using the Fourier-Bessel integral. This is 
the method followed by Lamb [L.3], p. 429. Instead of this procedure, 



160 WATER WAVES 

one could make use of the Hankel transform in a fashion exactly anal- 
ogous to the Fourier transform procedure used above (cf. Sneddon 
[S.ll], p. 290, and Hinze [H.15]). We content ourselves here with 
giving the result for the velocity potential 0(r, y; t) and the surface 
elevation r](r; t) due to the application of a concentrated unit impulse 
at the origin at t = 0: 

i r°° — 

(6.4.22) 0(r,y;t) = — — - \ e s yJ {sr) cos (Vgst)s ds, 

— 1 f 00 - 

(6.4.23) rj(r; t) = lim e sy J (sr) sin ( Vgs t)s m ds. 

27ZQ\/g y^oJO 

Naturally we want to discuss the character of the motions furnished 
by the above relations, and in doing so we come upon a fact that holds 
good in all problems of this type: it is a comparatively straightforward 
matter to obtain an integral representation for the solution, but not 
always an easy matter to carry out the details of the discussion of its 
properties. The reason for this is not far to seek — it is due to the fact 
that the solutions are given in terms of an integral over an integrand 
which is oscillatory in character and which changes rather rapidly 
over even small intervals of the integration variable for important 
ranges in the values of the independent variables. Hence even a nu- 
merical integration would not be easy to carry out. The fact is that the 
motions are really of a complicated nature, as we shall see, and hence 
a mathematical description of them can be expected to present some 
difficulties. Indeed, the phenomena under consideration here are 
analogous to the refraction and diffraction phenomena of physical 
optics and thus depend on intricate interference effects, which are 
further complicated in the present instances by the fact that the wave 
motions are subject to dispersion, as we have seen in Chapter 3. 

Some insight into the nature of the solutions furnished by our for- 
mulas can be obtained by expanding the integrands in power series 
and integrating term by term (cf. Lamb. [L.3], p. 385).* The result 
for rj(x; t) as given by (6.4.20), for example, is found to be (for x > 0): 



(6.4.24) rj(x; t) 



tcx[2x 1 • 3 • 5 \2x) 1 • 3 • 5 • 7 • 9 \2x) '"}' 

It is clear that there is a singularity for x = 0, as one would expect. 

* The subsequent discussion in this section follows closely the presentation 
given by Lamb. 



UNSTEADY MOTIONS 161 

The series converges for all values of the dimensionless quantity 
gt 2 2x, but practically the series is useful only for small values of 
gt 2 /2x, i.e. for small values of t, or large values of x. One observes also 
that any particular "phase" of the disturbance — such as a zero of rj, 
for example— must propagate with a constant acceleration, since any 
such phase is clearly associated with a specific constant value of the 
quantity gt 2 j2x. 

It is in many respects more useful to find an asymptotic represen- 
tation for the motion valid in the present case for large values of 
the quantity gt 2 /2x, for which the power series are not very useful 
because of their slow convergence. Indeed, the asymptotic represen- 
tation yields all of the qualitative features contained in the exact 
solution (6.4.24), and is also accurate even for rather small values of 
the quantity gt 2 /2x (cf. Sneddon [S.ll], p. 287). For this purpose it 
happens to be rather easy to work out an asymptotic development of 
the solution that is valid for large values of gt 2 /2x, and this we proceed 
to do, following Lamb. We write (6.4.19) in the form 

— Iff 00 — la 2 x \ 

(6.4.25) 0(x, y;t) = e<> * sin — + ot\ da 

r ° 2 y i^ x \ \ 

— e g sin I — — ot\ do ) , 

making use of or = Vgs, 2oda = gds. New quantities £ and a> are 
introduced in (6.4.25) by the relations 



from which 






G'X 






The expression (6.4.25) is thus readily found to take the form 

2 gl/2 /.« 

(6.4.26) 0(x, 0; t) = — ^ sin (£ 2 - oj 2 ) d£ 

7LX Jo 

where £ is introduced as new variable of integration and y is assumed 
to vanish. The corresponding free surface elevation is given by 



162 



WATER WAVES 



(6.4.27) 



--** 






f 

Jo 



cos (| 2 — co 2 ) d£ 



as one readily verifies. In order to study the last expression we con- 
sider the integral 



Jo Jo 



(6.4.28 

It is well known that 

r*oo 

(6.4.29) J e&~<»*) d g 



,i (l 2 -0> 2 ) 



d£ 



/»00 
J (O 



dS. 



y/n e 



while the second contribution can be treated as follows: 

1 P l 



(6.4.30) 



f 

•/a 



« 



If"! , 
2J w2 V* 

i r °° l r°° s 

& L a, 2 2 Ja> 2 



through introduction of 2 = £ 2 as new variable, and an integration 
by parts. We show next that the final integral is of the order co -1 . 
as follows: 



f 



t — 2e i(t-a>*) dt 



< il f-« g<(*-"") 



/»Q0 

Jo> 2 



tfr 



r i dfc = 2ft)" 1 . 



Upon considering the real parts of (6.4.28), (6.4.29), (6.4.30), and 
inserting in (6.4.27) one finds 

(Mjn , , ( .. t) = JL (g)i [ cos (£-=) + 0(co->) 

in which the function 0(ft> -1 ) refers, as one readily verifies, to a term 
which behaves like (gt 2 /4<x)~ 112 . Consequently, if co 2 = gt 2 j^x is suffi- 
ciently large, we may assume for the free surface elevation due to a 
concentrated surface elevation at x = and t = the approximate 
expression 



UNSTEADY MOTIONS 163 

<^- 32 > ^^(S) 4cos (£-i)- 

By continuing the integration by parts, as in (6.4.30), it would be 
possible to obtain approximations valid up to any order in the quanti- 
ty (o~ 1 = (gt 2 /4>x)~ 112 , but such an expansion would not be convergent; 
it is rather an asymptotic expansion correct within a certain order in 
a)- 1 when an appropriate finite number of terms in the expansion is 
taken. Expansions of this type are — as in other branches of mathe- 
matical physics— very useful in many of our problems and we shall 
have many other occasions to employ them. 

The case of a concentrated point impulse applied at x = at the 
time t = can be treated in exactly the same manner as the case 
just considered: one has only to begin with the solution (6.4.17) in- 
stead of (6.4.19), and proceed along similar lines. In particular, the 
approximate solution valid (to the same order in co" 1 ) for large values 
of gt 2 /4<x can be obtained; the result for the free surface elevation is 

- 2 /g* 2 \ 3/2 . Igt 2 n\ 
(6.4.33) ri(x;t)~ — sin |°- _ - . 

v ' /v ' Qgxt^7i\4xJ \4m 4/ 

The method used to derive these asymptotic formulas is rather 
special: it cannot be very easily used to study the cylindrical waves 
given by (6.4.23), for example. We turn, therefore, in the next section 
to the derivation of asymptotic approximations in all of these cases 
by the application of Kelvin's method of stationary phase. After- 
wards, the motions themselves will be discussed in section 6.6 on the 
basis of the approximate formulas. 

6.5. Application of Kelvin's method of stationary phase. 

The integrals of section 6.4 can all be put into the form 

(6.5.1) I(k) = J a %(£> k)e ik< ?® d£ 

without much difficulty, and this is a form peculiarly suited to an 
approximate treatment valid for large values of the real constant k. 
In fact, Kelvin seems to have been led to the approximate method 
known as the method of stationary phase through his interest in 
problems concerning gravity waves, in particular the ship wave 
problem. The general idea of the method of approximation is as fol- 



164 WATER WAVES . 

lows. When k is large the function exp {ik<p(g)} oscillates rapidly as £ 
changes, unless (p(g) is nearly constant, so that the positive and ne- 
gative contributions to the value of I(k) largely cancel out, provided 
that y)(i, k) is not a rapidly oscillating function of | when k is large. 
Hence one might expect the largest contributions to the integral to 
arise from the neighborhoods of those points in the interval from a 
to b at which cp(t;), the phase of the oscillatory part of the integral, 
varies most slowly, i.e., from neighborhoods of the points where 
cp'(^) = 0. This indeed turns out to be the case. In section 6.8 it will 
be shown that 

(6.5.2) /(*) = 2 W (. r ,k)[ 1 ^^ ^[i(k 9M ±^\ 

^ r(\) ( 6 u / i \ 

+ I *>(«.,*) ~^ ^777^) exp {ihp («,)} + [j^j . 

ByO(]//c 2/3 ) we mean a function which tends to zero like l/k 2/3 as k -> oo. 
In these expressions the sums are taken over all the zeros <x r of tp' (£) 
in the interior of the interval a 5^ | ^ b at which (p"{ct. r ) ^ and over 
the zeros a s of (p'(£) at which (p"(oL s ) = but (p"'(<x s ) ^ 0. The sign 
of the term i tt/4 in the first sum should be taken to agree with the 
sign of (p"(oi r ). The relation (6.5.2) is valid if ip(^, k) and cp(^) are ana- 
lytic functions of £ in a ^ £ ^ b, and if the only stationary points of 
<p(£) are such that q)"(£) and cp'"(^) do not vanish simultaneously.* 
We proceed to obtain the approximate solution (6.4.32) obtained 
in the previous section once more by this method. The motion of the 
water was to be determined for the case of an elevation of the water 
surface concentrated at a point; the formula for the velocity potential 
was put in the form (cf. (6.4.25)): 

1 f f 00 tl. (o 2 x \ 

(6.5.3) 0(x, y; t) = — - \\ e ^ sin I — + ot \ do 

f 00 °ll (o 2 x \ ] 

— e 9 sin I — — ot] do \ > 

Jo \ g / j 

* If a zero of <p'{£) of still higher order should occur, then terms of other types 
would appear, and the error would die out less rapidly in A;. It should also be 
noted that the coefficient function ip of section 6.8 is assumed to be independent 
of A;, which is not true in some of the examples to follow. However, it is not 
difficult to see that the proof of section 6.8 can be modified quite easily in such 
a way as to include all of our cases. 



UNSTEADY MOTIONS 165 

This can in turn be put in the form 

(6.5 A) 0(x, y; t) = e m * e i[mx+at) do — e m y e i{mx - ot) da 

with m = a 2 'g. It is understood that the imaginary part only is to be 
taken at the end. It is convenient to introduce a new dimensionless 
variable of integration as follows: 

2x 

(6.5.5) ^ = Jt a ' 

in terms of which (6.5.4) is readily found to take the form 

(6.5.6) 0(x, y;t) = -—[\ e™y c tk & +2S > d£ - \ e m * e ik ^~ 2 ^ d£ 

with 

gt 2 

(6.5.7) k = — 

as a dimensionless parameter. The quantity m is of course also a 
function of |, and exp {m(^)y} plays the role of the function y>(|) in 
(6.5.1). When the parameter k is large, we may approximate the 
solution by using (6.5.2). For the phases (p(tj) we have 

(6.5.8) tp(l) = P±2( 
with stationary points given by 

(6.5.9) p'Cf) = 2|±2 = 0, 

and we see that | = 1 is the only such point in the interval < | < oo 
over which the integrals are taken. Consequently only the second inte- 
gral in (6.5.6) possesses a point of stationary phase, and at this point 
we have 

(6.5.10) 9>"(1) = 2, cp(l)=-l. 

We obtain therefore from (6.5.2) the approximate formula 

(6.5.11 ) 0(x, y; t) ~ V H- g*»d)v A'^t) , 

' 71X 

as one readily verifies, and this formula is a good approximation for 
large values oik = gt 2 /4x. We can also calculate the free surface eleva- 



166 WATER WAVES 

tion r\ in the same way from rj = — (l/g)0 t | y=0 ; the result is easily 
found to be 



(6.5.12) ri(x;t)~ - /2 cos ( 



g ll2 t ig?_ 

2 v^ 3/2 cos W 



% 



just as before (cf. (6.4.32)). 

For the case of a concentrated impulse the method of stationary 
phase as applied to (6.4.17) or (6.4.18) leads to the following approxi- 
mation valid once again for large values of gt 2 /4<x: 



(6.5.13) rj(x; t] 



g H2 t 2 

4>\/7lQX 512 



. (gt 2 7t\ 

;in — — —I, 

\4aj 4/ 



and this coincides with the result given in (6.4.33). 

In the case of an impulse distributed over a segment one obtains 
from (6.4.16) the result 

2P . gt 2 a . (gt 2 7i\ 

(6.5.14) „(.; t) = - -j-^^ S1 „ - S,„ (- - -), 

valid for large values of gt 2 /4<uc.* 

For the ring waves furnished by (6.4.23) the asymptotic formula is 

gt 3 gt 2 

(6.5.15) rj(r; t) = — -=^ r sin — • 

To obtain this formula it is necessary to replace the Bessel function 
J (sr) in (6.4.23) by its integral representation 

'71 12 



2 C 71 ' 1 
Jo( sr ) = ~ cos ( sr cos P) dfi 



and then apply the method of stationary phase twice in succession. 
Since such a procedure is discussed later on in dealing with the 
simplified ship wave problem (cf. Chapter 8.1), we omit a discussion 
of it here, except to remark that the approximate formula (6.5.15) is 
valid for any r =£ and gt 2 \kr sufficiently large. 

* It may seem strange that this formula indicates that x = is a singular 
point for r\, while x = is not singular in the exact formula (6.4.16). This comes 
about through the introduction of the new variable (6.5.5) and the parameter A: 
in (6.5.7) which were used to convert the original integral to the form (6.5.1). 
However, the validity of the formula (6.5.2) is assured, as one can see from 
section 6.8, only if x ^ 0. 



UNSTEADY MOTIONS 167 

6.6. Discussion of the motion of the free surface due to disturbances 
initiated when the water is at rest 

We proceed to discuss the motions of the water surface in accord- 
ance with the results given in the preceding section. The general 
character of the motion is well given by the approximate formulas, 
and we shall therefore confine our discussion to them. We observe 
first that the oscillatory factors in the four approximate formulas 
(6.5.12) — (6.5.15) do not differ essentially, but the slowly varying 
nonoscillatory factors are different in the various cases: (a) at a fixed 
point on the water surface the disturbance increases in amplitude 
linearly in t in the case of an initial elevation concentrated at a point 
(cf. (6.5.12)), while for a fixed time the amplitude becomes large for 
small x like x~ 3/2 ; (b) in the case of an initial impulse concentrated 
at a point the amplitude increases quadratically in t at a fixed point, 
while for a fixed time the amplitude increases like x~ 5/2 for small 
x. (In these limit cases the approximate formulas are valid for x ^ 0, 
since the only other requirement is that the quantity gt 2 /4>x should be 
large.) The behavior of these solutions near x = is not very sur- 
prising since there is a singularity there. The behavior at any fixed 
point x as t -> oo is, however, somewhat startling: the amplitude is 
seen to grow large without limit as the time increases in both of these 
cases. This rather unrealistic result is a consequence of the fact that 
the singularity at the origin is very strong. If the initial disturbance 
were finite and spread over an area, the amplitude of the resulting 
motion would always remain bounded with increasing time, as one 
could show by an appeal to the general behavior of Fourier trans- 
forms.* This fact is well shown in the special case of a distributed 
impulse, as we see from (6.5.14), which is valid for all x 7^ and large 
t: the amplitude remains bounded as t -> 00. 

The general character of the waves generated by a point disturbance 
is indicated schematically in the accompanying figures which show 
the variation in surface elevation at a fixed point x when the time in- 
creases, and at a fixed time for all x. These figures are based on the 
formula (6.5.12) for the case of an initial elevation; the results for the 
case of initial impulse would be of the same general nature. 

* It is also a curious fact that the motion given by (6.5.14) for the case of 
an impulse over a segment requires infinite energy input, since the amplitude 
at any fixed point does not tend to zero . For the case of an initial elevation confined 
to a segment, however, the wave amplitude would die out with increasing time. 



168 



WATER WAVES 



It is worth while to discuss the character of the motion furnished 
by (6.5.12) in still more detail. It has already been remarked that any 
particular phase — such as a zero, or a maximum or minimum of rj — is 
of necessity propagated with an acceleration since each such phase is 
associated with a particular constant value of the quantity gt 2 /4<x: if 
the phase is fixed by setting gt 2 j^x = c, then this phase moves in 
accordance with the relation x = gt 2 /4c. The formula (6.5.12) holds 
only where the quantity gt 2 /4ix is large, and hence the individual pha- 
ses are accelerated slowly in the region of validity of this formula; or, 




Fig. 6.6.1a,b Propagation of waves due to an initial elevation 



UNSTEADY MOTIONS 169 

in other words, the phases move in such regions at nearly constant 
velocity. Also, for not too great changes in x or t the waves behave 
very nearly like simple harmonic waves of a certain fixed period and 
wave length. This can be seen as follows. Suppose that we vary t 
alone from t = t to t = t + At. We may write for the phase cp: 



4^o 



2At 



(0 






as one readily verifies. Thus if At/t is small, i.e. if the change At in the 
time is small compared with the total lapse of time since the motion 
was initiated, we have for the change in phase: 

Consequently the period T = At of the motion corresponding to the 
change Acp = 2ji in the phase is given approximately by the formula 

(6.6.1) T~ -. 

The accuracy of this formula is good, as we know, if T/t £± 4t7ix Q jgt% 
is small, and this is the case since gt q/4>x is always assumed to be large. 
Thus the period at any fixed point varies slowly in the time. In the 
same way one finds for the local wave length X the approximate 
formula 

$>7ix\ 

(6.6.2) A-i— p 

by varying with respect to x alone, and this is also easily seen to be 
accurate if gt\\kx^ is large. Thus for a fixed position x the period and 
wave length both vary slowly, and they decrease as the time increases, 
while for a fixed time the same quantities increase with x, as is borne 
out by the figures shown above. 

It is of considerable interest next to compute the local phase velo- 
city—the velocity of a zero of tj, for example— from gt 2 /4x = c 
when x and t vary independently; the result is 

dx 2x 

(6.6.3) — = — 

dt t 

for the velocity of any phase; thus for fixed x the phases move more 



170 WATER WAVES 

slowly as the time increases, but for fixed t more rapidly as x increases 
— that is, the waves farther away from the source of the disturbance 
move more rapidly, and they are also longer, as we know from (6.6.2). 
The wave pattern is thus drawn out continually, and the waves as 
they travel outward become longer and move faster. The last fact is 
not too surprising since the waves in the vicinity of a particular point 
have essentially the simple character of the sine or cosine waves of 
fixed period that we have studied earlier, and such waves, as we have 
seen in Chapter 3, propagate with speeds that increase with the wave 
length. All of the above phenomena can be observed as the result of 
throwing a stone into a pond; though the motion in this case is three- 
dimensional it is qualitatively the same, as one can see by comparing 
(6.5.15) with (6.5.12). 

There is another way of looking at the whole matter which is 
prompted by the last observations. Apparently, the disturbance at 
the origin acts like a source which emits waves of all wave lengths and 
frequencies. But since our medium is a dispersive medium in which 
the propagation speed of a particular phase increases with its wave 
length, it follows that the disturbance as a whole tends with increasing 
time to break up into separate trains of waves each of which has ap- 
proximately the same wave length, since waves whose lengths differ 
move with different velocities. However, it would be a mistake to 
think that such wave trains or groups of waves themselves move with 
the phase speed corresponding to the wave length associated with the 
group. If one fixes attention on the group as a whole rather than on 
an individual wave of the group, the velocity of the group will be seen 
to differ from that of its component waves. The phase velocity for the 
present case can be obtained in terms of the local wave length readily 
from the equation (6.6.3) by expressing its right hand side in terms 
of the local wave length through use of (6.6.2); the result is 



(6.6.4; 



dx 2x _ i Igi 
dt t r 2^7 



On the other hand, the position x of a group of waves of fixed wave 
length X at time t is given closely by the formula 

(6.6.5) x = \]/&t, 

as we see directly from (6.6.2), so that the velocity of the group is 



UNSTEADY MOTIONS 171 



|VgA/27c, which is, evidently, just half the phase speed of its com- 
ponent waves. In other words, the component waves in a particular 
group move forward through the group with a speed twice that of the 
group. 

Finally, we observe that these results are in perfect accord with the 
discussion in Chapter 3 concerning the notions of phase and group 
velocity. The phase speed c for a simple harmonic wave of wave length 
X in water of infinite depth is given (cf. (3.2.3)!), by c = VgX/2ji, 
and this is also the phase speed of the waves whose wave length is 
X — as we see from (6.6.4). We have also defined in section 3.4 the 
notion of group velocity for simple harmonic waves in water of infinite 
depth, and found it to be just half the phase velocity. The kinematic 
definition of the group velocity given in section 3.4 was obtained by 
the superposition of trains of simple harmonic waves of slightly differ- 
ent wave length and amplitude, while in the present case the waves 
are the result of a superposition of waves of all wave lengths and 
periods. However, the principle of stationary phase, which furnishes 
the approximate solution studied here, in effect says that the main 
motion in certain regions is the result of the superposition of waves 
whose wave lengths and amplitudes differ arbitrarily little from a 
certain given value. The results of the analysis in the present case are 
thus entirely consistent with the analysis of section 3.4. 

At any time, therefore, the surface of the water is covered by groups 
of waves arranged so that the groups having waves of greater length 
are farther away from the source. These groups, therefore, tend to 
separate, as one sees from (6.6.4). The waves in a given group do not 
maintain their amplitude, however, as the group proceeds: one sees 
readily from (6.5.12) in combination with (6.6.2) that their amplitude 
is proportional to \j\/x for waves of fixed length X. 

The above interpretations of the results of the basic theory are all 
borne out by experience. Figure 6.6.2 shows a time sequence of photo- 
graphs of waves (given to the author by Prof. J. W. Johnson of the 
University of California at Berkeley) created by a disturbance 
concentrated in a small area: the decrease in wave length at a fixed 
point with increasing time, the increase in the wave lengths near the 
front of the outgoing disturbance as the time increases, the general 
drawing out of the wave pattern with time, the occurrence of well- 
defined groups, etc. are well depicted. 

An interesting development in oceanography has been based on the 
theory developed in the present section. Deacon [D. 6, 7] and his 



172 



WATER WAVES 




Fig. 6.6.2. Waves due to a concentrated disturbance 



UNSTEADY MOTIONS 



173 




Fig. 6.6.2. (Continued) 



174 WATER WAVES 

associates have carried out studies which correlate the occurrence of 
storms in the Atlantic with the long waves which move out from the 
storm areas and reach the coast of Cornwall in a relatively short time. 
By analyzing the periods of the swell, as determined from actual 
wave records, it has been possible to identify the swell as having been 
caused by storms whose location is known from meteorological obser- 
vations. Aside from the interest of researches of this kind from the 
purely scientific point of view, it is clear that such hindcasts could, in 
principle, be turned into methods of forecasting the course of storms 
at sea in areas lacking meteorological observations. 

6.7. Waves due to a periodic impulse applied to the water when 
initially at rest. Derivation of the radiation condition for purely 
I periodic waves 

In section 3 of Chapter 4 we have solved the problem of two-dimen- 
sional waves in an infinite ocean when the motion was a simple 
harmonic motion in the time that was maintained by an application 
of a pressure at the surface which was also simple harmonic in the 
time. In doing so, we were forced to prescribe radiation conditions 
at oo — effectively, conditions requiring the waves to behave like out- 
going progressing waves at oo — in order to have a complete formula- 
tion of the problem with a uniquely determined solution. It was 
remarked at the time that a different approach to the problem would 
be discussed later on which would require the imposition of bounded- 
ness conditions alone at oo, rather than the much more specific radia- 
tion condition. In this section we shall obtain the solution worked out 
in 4.3 without imposing a radiation condition by considering it as the 
limit of an unsteady motion as the time tends to infinity. However, it 
has a certain interest to make a few remarks about the question of 
radiation conditions in unbounded domains from a more general point 
of view (cf. [S. 21]). 

In wave propagation problems for what will be called here, ex- 
ceptionally, the steady state, i.e., a motion that is simple harmonic in 
the time, it is in general not possible to characterize uniquely the 
solutions having the desired physical characteristics by imposing only 
boundedness conditions at infinity. It is, in fact, as we have seen in 
special cases, necessary to impose sharper conditions. In the simplest 
case in which the medium is such as to include a full neighborhood of 
the point at infinity that is in addition made up of homogeneous matter, 



UNSTEADY MOTIONS 175 

the correct radiation condition is not difficult to guess. It is simply 
that the wave at infinity behaves like an outgoing spherical wave 
from an oscillatory point source, and such a condition is what is 
commonly called the radiation, or Sommerfeld, condition. Among 
other things this condition precludes the possibility that there might 
be an incoming wave generated at infinity — which, if not ruled out, 
would manifestly make a unique solution of the problem impossible. 

If the refracting or reflecting obstacles to the propagation of waves 
happen to extend to infinity— for example, if a rigid reflecting wall 
should happen to go to infinity — it is by no means clear a priori what 
conditions should be imposed at infinity in order to ensure the unique- 
ness of a simple harmonic solution having appropriate properties 
otherwise.* A point of view which seems to the author reasonable is 
that the difficulty arises because the problem of determining simple 
harmonic motions is an unnatural problem in mechanics. One should in 
principle rather formulate and solve an initial value problem by 
assuming the medium to be originally at rest everywhere outside a 
sufficiently large sphere, say, and also assume that the periodic 
disturbances are applied at the initial instant and then maintained 
with a fixed frequency. As the time goes to infinity the solution of the 
initial value problem will tend to the desired steady state solution 
without the necessity to impose any but boundedness conditions at 
infinity.** 

The steady state problem is unnatural — in the author's view, at 
least — because a hypothesis is made about the motion that holds 
for all time, while Newtonian mechanics is basically concerned with 
the prediction — in a unique way, furthermore — of the motion of a 
mechanical system from given initial conditions. Of course, in me- 
chanics of continua that are unbounded it is necessary to impose con- 
ditions at oo not derivable directly from Newton's laws, but for the 
initial value problem it should suffice to impose only boundedness 
conditions at infinity. In sec. 6.9. the relevant uniqueness theorem for 
the special case to be considered later is proved. 

* For a treatment of the radiation condition in such cases see Rellich [R.7], 
John [J.5], and Chapter 5.5. 

** The formulation of the usual radiation condition is doubtlessly motivated 
by an instinctive consideration of the same sort of hypothesis combined with the 
feeling that a homogeneous medium at infinity will have no power to reflect 
anything back to the finite region. Evidently, we also have in mind here only 
cases in which no free oscillations having finite energy occur — if such modes of 
oscillation existed, clearly no uniqueness theorems of the type we have in mind 
could be derived. 



176 WATER WAVES 

If one wished to be daring one might, on the basis of these remarks, 
formulate the following general method of obtaining the appropriate 
radiation condition: Consider any convenient problem in which the 
part of the domain outside a large sphere is maintained intact and 
initially at rest. (In other words, one might feel free to modify in any 
convenient way any bounded part of the medium.) Next solve the 
initial value problem for an oscillatory point source placed at any 
convenient point. Afterwards a passage to the limit should be made in 
allowing the time t to approach oo, and after that the space variables 
should be allowed to approach infinity. The behavior at the far distant 
portions of the domain should then furnish the appropriate radiation 
conditions independent of the constitution of the finite part of the 
domain. It might be worth pointing out specifically that this is a case 
in which the order of the two limit processes cannot be interchanged: 
obviously, if the time t is first held fixed while the space variables tend 
to infinity the result would be that the motion would vanish at oo, 
and no radiation conditions could be obtained. 

The writer would not have set down these remarks — which are of a 
character so obvious that they must also have occurred to many 
others — if it were not for two considerations. Every reader will doubt- 
lessly have said to himself: "That is all very well in principle, but will 
it not be prohibitively difficult to carry out the solution of the initial 
value problem and to make the subsequent passages to the limit?" 
[n general, such misgivings are probably all too well founded. How- 
ever, the problem concerning water waves to be treated here happens 
to be an interesting special case in which (1) the indicated program 
can be carried out in all detail, and (2) it is slightly easier to solve the 
initial value problem than it is to solve the steady state problem with 
the Sommerfeld condition imposed. 

We restrict ourselves to two-dimensional motion in an x, y-p\ane, 
with the y-axis taken vertically upward and the «-axis in the originally 
undisturbed horizontal free surface. The velocity potential <p(x, y; t) 
is a harmonic function in the lower half-plane: 

(6.7.1) <p<c X + Vyy = 0, y<0, t>0. 

The free surface boundary conditions are (cf. (6.1.1), (6.1.2)): 



for y = 0, t > 0. 



(6.7.2) 


~<Pv +Vt 


= 


(6.7.3) 


<Pt +gy = - 


i 

--v 
p 



UNSTEADY MOTIONS 177 

As usual, y\ = rj(x; t) represents the vertical displacement of the free 
surface measured from the £-axis, and p = p(x; t) represents the 
pressure applied on the free surface. We suppose that <p and its first 
and second derivatives tend to zero at oo for any given time t— in fact 
that they tend to zero in such a way that Fourier transforms exist — 
but we do not, in accordance with our discussion above, make any 
more specific assumptions about the behavior of our functions as 
t -» oo. At the time t = we prescribe the following initial conditions 

(6.7.4) <p(x, 0; 0) = <p t (x 9 0; 0) = 0, 

which state (cf. (6.1.6), (6.1.7)) that the free surface is initially at 
rest in its horizontal equilibrium position. 

In what follows we consider only the special case in which the sur- 
face pressure p(cc; t) is given by 

(6.7.5) p(x; t) = d(x)e itot , t >0 

in which d(cc) is the Dirac ^-function. We have not made explicit use 
of the (3-function until now, but we have used it implicitly in section 
6.4 in dealing with concentrated impulses. It is to be interpreted in 
the same way here, i.e. as a symbol for a limit process in which the 
pressure is first distributed over a segment the length of which is 
considered to grow small while the total pressure is maintained at the 
constant value one. By inserting this expression for p in (6.7.3) and 
eliminating the quantity r\ by making use of (6.7.2) the free surface 
condition is obtained in the form 

(6-7.6) g<p v +<p u = d(x)e tat , t > 0. 

Our problem now consists in finding a solution <p(x, y; t) of (6.7.1) 
which behaves properly at oo, and which satisfies the free surface 
condition (6.7.6) and the initial conditions (6.7.4). 

We proceed to solve the initial value problem by making use of the 
Fourier transform applied to the variable x. The result of transforming 
(6.7.1) is 

(6.7.7) - s*y + $ yy = 0, 

in which (p(s, y; t) is the transform of cp(x, y; t) and use has been made 
of the conditions at oo. The bounded solutions of (6.7.7) for y < 0, 
* > are all of the form 



178 WATER WAVES 

(6.7.8) $(s,y;t) = A(s; 

The transform is now applied to the boundary condition (6.7.6), 
with the result: 

1 ico 

(6.7.9) gcp y +<p tt = - —= — e™\ for y = 0, 

and on substitution of <p(s, 0; t) from (6.7.8) we find 

1 ico 

(6.7.10) A u + gsA = - —=. — e™K 

\/2ti q 

The initial conditions (6.7.4) now furnish for A (s; t) the conditions 

(6.7.11) A(s; 0) = A t {s; 0) = 0. 

The solution of (6.7.10) subject to the initial conditions (6.7.11) is 

1 ico f ' e ia> ^~ T) /— 

(6.7.12) A(s; t) = = — =- sin Vgs rdr. 

V2jz Q Jo Vgs 



Finally, we insert the last expression for A (s; t) in (6.7.8) and apply 
the inverse transform to obtain the following integral representation 
for our solution cp(x, y; t): 

(6.7.13) <p(x, y;t)=— — e sy cos sx =— sin Vgs rdrds. 

9^ Jo Jo Vgs 

The fact that the solution is an even function of x has been used here. 
Our object now is to study the behavior of this solution as t -» oo. 
Since y is negative (we do not discuss here the limit as y -> 0, 
i.e. the behavior on the free surface) the integral with respect to s 
converges well and there is no singularity on the positive real axis of 
the complex s-plane. However, the passage to the limit t -> oo is more 
readily carried out by writing the solution in a different form in 
which a singularity — a pole, in fact — then appears on the real axis 
of the s-plane. (It seems, indeed, likely that such an occurrence would 
be the rule in any considerations of the present kind since the limit 
function as t -> oo would not usually be a function having a Fourier 
transform, and one could expect that the limit function would some- 
how appear as a contribution in the form of a residue at a pole.) It is 



UNSTEADY MOTIONS 



179 



convenient to deform the path of integration in the s-plane into the 
path L indicated in the accompanying figure. The path L lies on the 



s - plane 



r\ 



Fig. 6.7.1. Path of integration in s-plane 

positive real axis except for a semicircle in the upper half-plane cen- 
tered at the point s = co 2 /g. By Cauchy's integral theorem this leaves 
the function 99 given in (6.7.13) unchanged. 

We now replace sin Vgs r in (6.7.13) by exponentials and carry 
out the integration on x to obtain 



icoe' 



L 




> ds. 



(6.7.14) <p(x, y;t) = — I e sy cos sx { 

KQ Jl 



We wish now to consider the three items in the bracket separately, 
and, as we see, two of them do indeed have a singularity at s = w 2 /g 
which is by-passed through our choice of the path L. The first two 
items are rather obviously the result of the initial conditions and 
hence could be expected to pro\ide transients which die out as 
t -> 00. This is in fact the case, as can be seen easily in the following 
way: That branch of \/s is taken which is positive on the positive real 
axis, and we operate always in the right half-plane. If, in addition, 
s is in the upper half-plane it follows that ii^gs ± co) has its real part 
negative (00 being real). Consider now the contribution furnished by 
the uppermost item in the square brackets. Since the exponential has 
a negative real part on the semi-circular portion of the path L it is 
clear that as t -> + go this part of the path makes a contribution 
that tends to zero. The remaining portions of L, which lie on the real 



180 WATER WAVES 

axis, are then readily seen to make contributions which die out 
like 1//: this can be seen easily by integration by parts, for example, 
or by application of known results about Fourier transforms. The 
middle item in the square brackets has no singularity on the real 
axis, so that the path L can be taken entirely on the real axis; thus, 
in accordance with the remarks just made concerning the similar 
situation for the first item, it is clear that this contribution also dies 
out like 1//. Thus for large t we obtain the following asymptotic re- 
presentation for op: 



ico . . C e sy cos sx , 

(6.7.15) <p(x, y; t) ~ — — e imt ds. 

J L gs - CO 2 



71Q 



Actually, the right hand side is the solution of the steady state 
problem — as obtained, for example, in the paper of Lamb [L.2] and 
by a different method by us in section 4.3 (although in a different form) 
— when the condition at oo is the radiation condition stating that cp 
behaves like an out-going progressing wave. The steady state solution 
as obtained in section 4.3 actually was a little more awkward to obtain 
directly through use of the radiation condition than it was to obtain 
the solution (6.7.13) of the initial value problem. In particular, the 
asymptotic behavior of an integral representation had to be investi- 
gated in the former case also before the radiation condition could be 
used. Thus we have seen in this special case that the radiation condi- 
tion can be replaced by boundedness conditions (in the space varia- 
bles, that is) if one treats an appropriate initial value problem instead 
of the steady state problem. 

Even though not strictly necessary — since (6.7.15) is known to 
furnish the desired steady state solution — it is perhaps of interest to 
show directly that the right hand side of (6.7.15) has the behavior 
one expects for an out-going progressing wave when x -> + 00. The 
procedure is the same as that used in discussing (6.7.14): The factor 
cos sx is replaced by exponentials to obtain 



gsy p—isx 

ds 

gS — (JO 2 



co r 1 f e sy e isx 1 C 

(6.7.16) cp(x.y;t)^-e^ — d* + — 

9 \J2wjLgs — co 2 ZjiiJl 

By the same argument as above one sees that the first integral makes 
a contribution that tends to zero as x -> + 00. The second integral 
is treated by deforming the path L over the pole s = co 2 /g into a path 
M which consists of the positive real axis except for a semi-circle 
in the lower half-plane. The contribution of the second integral then 



UNSTEADY MOTIONS 181 

consists of the residue at the pole plus the integral over the path M. 
But the contribution of the latter integral is, once more, seen to tend 
to zero as x -> + °o because of the factor e~ isx . Thus <p(x, y; t) behaves 
for large x as follows: 

CO 2 /CO 2 \ 

Q) —y —il —x—(ot ) 

(6.7.17) <p(x, y)~ — — e 9 e Kg J , 



and this does in fact represent a progressing wave in the positive 
^-direction which, in addition, has the wavelength 2jrg/co 2 appropriate 
to a progressing sine wave with the frequency oj in water of infinite 
depth. 

6.8. Justification of the Method of Stationary Phase 

In section 6.5 the method of stationary phase was used (and it will 
be used again later on) to obtain approximations of an asymptotic 
character for the solutions of a variety of problems when these 
solutions are given by means of integrals of the form 

(6.8.1) 1(h) = j ip(x)e ik ^ dx, 



and the object is to obtain an approximation valid when the real 
constant k is large. Since we make use of such approximate formulas 
in so many important cases, it seems worth while to give a mathema- 
tical justification of the method of stationary phase, following a pro- 
cedure due to Poincare. The presentation given here is based upon the 
presentation given by Copson [C.5]. 

Poincare's proof requires the assumption that (p(z) and ip(z) are 
regular analytic functions of the complex variable z in a domain 
containing the segment S: a ^ x ^ b of the real axis in its interior. 
(In what follows, we assume a and b to be finite, but an extension to 
the case of infinite limits would not be difficult.) In addition cp(z) is 
assumed to be real when z is real. These conditions are more restric- 
tive than is necessary for the validity of the final result. For example, 
the function y> might also depend on k, provided that ip(x, k) is not 
strongly oscillatory, or singular, for large values of k. The assumption 
of analyticity is also not indispensable. However, these generaliza- 
tions would complicate both the formulation and proof of our theorem 
without changing their essentials; consequently we do not consider 
them here. 



182 WATER WAVES 

It will be shown that the main contributions to I(k) arise from the 
points of S near those values of x for which <p'(x) = 0— that is. near 
the points of stationary phase. The term of lowest order in the asymp- 
totic development of I(k) with respect to k will then be obtained on 
the basis of this observation. Kelvin himself offered a heuristic argu- 
ment (cf. sec. 5 above) indicating why such a procedure should yield 
the desired result. 

Since <p'(z) is regular in the domain containing S, it follows that its 
zeros are isolated. Hence S can be divided into a finite number of 
segments on which 99(2) has either one stationary point or no stationary 
point. We shall show first that the contribution to/(A;) from a segment 
containing no stationary point is of order 1/k. Next it will be shown 
that a segment containing any given point of stationary phase can be 
found such that the contribution to the integral furnished by the 
segment is of lower order than 1/k, and a formula for this contribution 
will be derived. It turns out that this contribution of lowest order 
is independent of the length of the segment containing the point 
of stationary phase, provided only that the segment has been chosen 
short enough. Once these facts have been proved, it is clear that the 
lowest order contributions to the integral are to be found by adding 
the contributions arising at each of the points of stationary phase. 

Suppose, then, that cp(x) has no stationary point on a segment 
c 5^ x ^ d of S. We may write 



h 



[ d W ( X ) e i«<p[*) dx = Tj^L i. ( C «*(.>) dx, 

J c J c ik(p'{x) dx 



since (p'(x) ^Oinc^^^dby hypothesis. Integration by parts 
then leads to the result 

I, = W{d) e ik *W _ 7M. g«*(e) _ A f %«*>! dx, 
ikcp'(d) ik(p'{c) z'A:J c 



I dx 



d ' C d C d 

with ip^x) = — (yl<p')- Since | e ik(pr ip x dx ^ 1^ 
doc ' I Jc Jc 

because of the fact that kcp(x) is real, it follows that the integral in the 
above expression is bounded. Thus I 1 is indeed of order 1/k, as stated 
above. It might be noted that this argument really does not require 
the analyticity of cp and \p, but only that the integrands be integrable 
and that integration by parts may be performed. Infinite limits for the 
integrals could also be permitted if (p(x) and xp(x) behave appropriately 
at 00. 



UNSTEADY MOTIONS 183 

Suppose now that cp(x) has one stationary value at x = a in the 
segment a — s 1 ^ x ^ a + e l9 e ± > 0, i.e.,, (p'(x) vanishes only at 
x = a in this interval. Suppose, in addition, that the second deriva- 
tive (p"(x) does not vanish at x = a, and indeed is positive there: 
(p"(oi) > 0. (The case in which 9?" (a) is negative and the more critical 
case in which 9?"(a) = will be discussed later.) We shall show that 
a positive number e ^ s x exists such that 

(6.8.2) I 2 (k) = f %(*)« rt * (s) dx = p^%( a )/^ (a)+ l) + (-). 

Ja-e W («)/ W 

In other words, we shall show that a fixed segment of length 2s con- 
taining a exists such that its contribution to / is independent of s and 
is of order l/\/k, with an error of order 1/k. 

To prove these statements we begin by introducing new variables 
as follows: 

(6.8.3) x = a + u, cp(x) = 99(a) + w(u). 
Consider first the integral I 2 (k, e x ): 

(6.8.4) I 2 {k, £ X ) = e ik(p{a) j * e ikw{u) <ip(o<. + u) du = e ik(p{a) J. 

It is convenient to write the integral J as the sum of two terms: 

(6.8.5) J = g<*"(«i) xp(ai + Ul ) du x + l g**«C».) y,( a _|_ Wg ) rfw 2 

= Ji + J* 
Since 99(37) has a minimum at a? = a, it follows that w^) is a positive 
monotonic function in the interval — e 1 5^ u x 5^ 0, and likewise 
za(w 2 ) in the interval ^ w 2 ^ £ x . Hence we may introduce a new 
integration variable t, which is furthermore real, in each of the in- 
tegrals, defined as follows: 

t 2 = «)(%) in — £ x ^ u x ^ 0, and 



(6.8.6; 

1 t 2 = w(u 2 ) in 5^ u 2 ^ e x . 

In each interval t is taken as the positive square root. The integrals 
J x and J 2 become, as one readily sees: 



(6.8.7; 



Jo « 

Jo a* 



184 WATER WAVES 



with t x = Vw(— £ x ), and t 2 — Vw(e 1 ). The functions u^t), u 2 (t) are 
solutions of w(Ui) = t 2 . For w(u) we have the power series develop- 
ment 

(6.8.8) w(u) = 9?(a + u) — 99(a) = a 9 u 2 -f « 3 w 3 + . . . 

since w(0) = w'(0) = (cf. (6.8.3)); in addition 2a 2 = (p"(<x.) > 0, 
by assumption. We suppose that this series converges in a circle which 
contains the entire interval — e 2 5^ u ^ s 2 in its interior, with 
e 2 < s v Since t 2 = w(u 2 ) we may write 



(6.8.9; 



u 2 Va 2 + «3^ 2 + • • • f° r ^ w 2 5^ So and 



£ = — u x \/ a 2 + fl^ + . . . for — e 2 ^ w x ^ 0. 

Since « 2 ^ we may express the square roots as power series in u t 
and then invert the series to obtain u x and u 2 as power series in t, as 
follows : 

1 = — c x t + c 2 t 2 + . . ., 



(6.8.10; 

c x £ + c 2 t 2 + 



with c x = + V 2/99" (a). Hence we may write 

oLUi 
- y>(oc + Uj) — * = c lV >(a) + tP^O, 

V(« + w 2 ) -37 = c iWM + ^P 2 (0» 

in which P^t) and P 2 (/) are convergent power series. It may be that 
these series do not converge up to the values t x and t 2 of the upper li- 
mits of the above integrals J 1 and J 2 in (6.8.7). In that case we simply 
assume the length of the segment is taken to be still less than 2e 2 so 
that the inversion of the series (6.8.8) is permissible and the series 
P-i(t) and P 2 (t) converge up to appropriate values i x and l 2 . It is 
clear that numbers i t and l 2 with these properties exist. The integrals 
J x and J 2 may now be written in the form* 



(6.8.11) 



«/i = f h e ikt * {wW + tP x (t)} dt, and 

J 2 = c lV ){<x) ( h e iki2 dt + ! h e ikt2 tP 2 (t)dt = J 3 + J 4 . 



* The requirement of analyticity for <p and xp is used to permit this simple 
introduction of t as variable of integration. However, the existence of a finite 
number of derivatives would clearlv have sufficed. 



UNSTEADY MOTIONS 



185 



We proceed to study the integrals J 3 and J 4 . Upon introducing 6 = Jet' 1 
as new variable in Jo we obtain 



%Vk Jo 



kti id 



VO 



dd. 



But we may write 
(6.8.12) 



fkq p tV fee ptV [*ao 

— dd = \ ~dd - 

Jo VO Jo VO j mi 



Vne T 



/»00 

Jul 



VO 



9 

dd, 



by a known formula. The last integral can now be shown to be of 
order l/\/A; by integration by parts, as follows: 



Jktl 



VO 



VO 



00 I /»Q0 ,,Z0 



d0 . 



The first contribution on the right hand side is obviously of order 
1/Vh since i 2 is a fixed number; as for the second, we have 



/»oo p id |*oo 

— - dd < 0- 3 ' 2 <$ 



2 

hV k 



"2 " ^ 

and hence the second contribution is also of order 1/\A- Thus for J 3 
we have the result 

(6.8.13) J 3 = fey(a) Q «*"* 1 + O (-U . 

The integral J 4 is first integrated by parts to obtain 

J 4 = ( * e ikt * tP 2 {t) dt 

Jo 



1 

2ik 



e™ 2 P 2 (l 2 ) - P 2 (0) 



-\: 



^ Pg(«) dz 



and hence 



(6.8.14) i j 4 1 <: 1 j | p f ft) | + | p 2 (o) i + f 2 1 p;(o | A 

and the right hand side is thus of order 1/Zc. The integral J 1 can ob- 
viously be treated in the same way as J 2 and with an exactly analogous 



186 WATER WAVES 

result; consequently we have from (6.8.13), and (6.8. 14) for the integral 
given in (6.8.4) the result 

(6.8.15) /.(ft, H ) = / 2 (fc) = V («) (~¥^X e< (**<«> + j) + O Q , 

once £ 3 has been chosen small enough. One observes how it comes 
about that the lowest order term is independent of the values of t x and 
£ 2 , and hence of the length of the segment: the entire argument re- 
quires only that i± and i 2 be any fixed positive numbers since one needs 
only the fact that the products kl\ and kl\ grow large with k. 

If (p(x) had been assumed to have a maximum at x = a, with 
<p" (a) < 0, the only difference would be that — k<p"(ct.) and — tt/4 
would appear in the final formula instead of + kq>"(a.) and -f jt/4. 
Consequently, in all cases in which cp" (a) ^ we have 

(6.8.16) / 2 (ft) = y(a) ( ** ) <■<**<«>*;) + O (J) 

\ft|y"(a)i; w 



,« r(i) 



and the sign of the term m/4 should agree with the sign of 9?" (a). 
Finally, in case 99" (a) = 0, but q>'" (a) 7^ it is not difficult to 
derive the appropriate asymptotic formula for I(k). In fact, the steps 
are nearly identical with those taken just now for the case cp" (a) ^ 0. 
One introduces x = a + u, cp(x) = 99(a) + w(u) as before and then 
makes use of power series in the variable t defined by t 3 = w(u) in the 
same way as above. The result is, for s sufficiently small: 

(6.8.17) I 2 (k) = e ik(p{x) yj(x) dx 

J a— e 

( - \*.^(«)+o(-V| 

V3 \W'(a)l/ U 2/8 / 

where /^(J) refers to the gamma function. Hence the contribution 
arising from the stationary point is now of a different order of magni- 
tude, i.e., of order 1/k 113 instead of l/k l12 . This fact is of significance 
in the case of the ship wave problem which will be treated later. 

Naturally the lowest order terms in I(k) consist of a sum of terms 
furnished by the contributions of all of the points of stationary phase 
in the interval S. It is important enough to bear repetition that if no 
such points exist, then I(k) is in general of order 1/k. 

In case a stationary point falls at an end point x = a or x = b of 
the interval of integration, one sees readily that the contribution 
furnished by such a point to I(k) is the same as that given above in 



UNSTEADY MOTIONS 187 

case 9?" ^ except that a factor 1/2 would appear in the final result. 
On the other hand, if op" = but g/" ^ at an end point, then the 
contribution differs in phase as well as in the numerical factor from 
the contribution given above in (6.8.17). 

6.9. A time-dependent Green's function. Uniqueness of unsteady 
motions in unbounded domains when obstacles are present 

In sec. 6.2 above the uniqueness of unsteady wave motions for 
water confined to a vessel of finite dimensions was proved. More gener- 
al results have been obtained by Kotik [K.17], Kampe de Feriet and 
Kotik [K.l], and Finkelstein [F.3] with regard to such uniqueness 
questions. In the present section a rather general uniqueness theorem 
will be proved, following the methods of Finkelstein, who, unlike the 
other authors mentioned, obtains uniqueness theorems when obstacles 
are present in the water. The essential tool for this purpose is a time- 
dependent Green's function, which is in itself of interest and worth 
while discussing for its own sake quite apart from its use in deriving 
uniqueness theorems. With the aid of such a function, for example, all 
of the problems solved in the preceding sections can be solved once 
more in a different fashion, and still other and more complicated un- 
solved problems can be reduced to solving an integral equation, as 
we shall see. 

We shall derive the time-dependent Green's function in question 
for the case of three-dimensional motion in water of infinite depth, 
although there would be no difficulty to obtain it in other cases as 
well. The Green's function G in question is required to be a harmonic 
function in the variables (x, y, z) with a singularity of appropriate 
character at a certain point (f, rj, £) which is introduced at the time 
t = t and maintained thereafter; thus G depends upon £, rj, f ; r and 
x, y, z; t: G = G(£, rj 9 f ; r \ x, y, z; t). In fact, G is the velocity poten- 
tial which yields the solution of the following water wave problem: 
A certain disturbance is initiated at the point (£, rj, £) at the time 
t = t. The pressure on the free surface of the water is assumed to be 
zero always, and at the time t = r the water is assumed to have been 
at rest in its equilibrium position. Since G is a harmonic function in 
x, y, z it is reasonable to expect that the correct singularity to impose 
at the point (|, rj, f ) in order that it should have the properties one 
likes a Green's function to have is that it behaves there like 1/R, with 
R - V(f - x) 2 + (tj - yf + (C - zf. Thus G should satisfy the 



188 WATER WAVES . 

following conditions: It should be a solution of the Laplace equation 

(6.9.1) G xx + G yy + G ZZ = for - oo < y < 0, t ^ r, 
satisfying the free surface condition 

(6.9.2) G tt +gG y = 0, y = 0. 

At oo we require G, G t and their first derivatives to be uniformly 
bounded at any given time t. (Actually, they will be seen to tend to 
zero at oo.) At the point £, r), £ we require 

(6.9.3) G — — to be bounded. 

R 

As initial conditions at the time t = t we have (cf. sec. 6.1) 

(6.9.4) G = G t = for t = t, y = 0. 

As we shall see later on, these conditions determine G uniquely. 
We proceed to construct the function G explicitly. As a first step 
we set 

(6.9.5) G(( t rj,t;r\ x, y, z; t) = A& rj, £ \ x, y, z) + 

B(£ ; t],£;t I x, y, z; t) 
with A defined by 



1 1 



(6.9.6) A=--— with R' = V(| - ^) 2 + (^ + 2/) 2 + (f - 2) 2 . 

Thus A contains the prescribed singularity, and we may require B 
to be regular. Since A is a harmonic function, it follows that B is 
harmonic; in addition, B satisfies the free surface condition 

(6.9.7) B tt +gB v = 



[(1 

2g 


- x) 2 

d 


+ y 2 


1 


- z) 2 ] 3 ' 2 




drjKi 


— x)' 


2 +^ 2 


+ (C- 


z )2jl/2 



at z/ = 0, 

as one can readily verify. To determine B from this and the other con- 
ditions arising from those imposed on G it would be possible to employ 
the Hankel transform in exactly the same way as the Fourier trans- 
form was used in preceding sections. However, it seems better in the 
present case to proceed directly by using the special, but well-known, 
Hankel transform for the function e~ bs /s (cf., for example, Sneddon 
[S.ll], p. 528); this yields the formula 



UNSTEADY MOTIONS 189 



1 r* 

(6.9.8) = e- bs J (as)ds, 

Va 2 + b 2 Jo 



Va 2 + b 2 

valid for b > 0. By means of this formula the right hand side of (6.9.7) 
can be written in a different form to yield 



(6.9.9) B tt + gB y = 2g - f V J (sr) ds = 2g 

tyJo 

valid for rj < and with 



se v * J (sr) ds 

at y = 



(6.9.10) r = V (| - «) 2 + (f — z) s 



Since J? is a harmonic function in x, y, z, it would seem reasonable to 
seek it among functions of the form 

/•Q0 

(6.9.11) B = sT(t, s)e {v ^ )s J (sr) ds, 



/•Q0 

Jo 



which are harmonic functions. The free surface condition (6.9.9) will 
now be satisfied, as one can easily see, if T(t) satisfies the differential 
equation 

(6.9.12) T tt +gsT = 2g. 

The function T is now uniquely determined from (6.9.12) and the 
initial conditions T = T t = for t = x derived from (6.9.4); the 
result is 

(6.9.13) Tit, s) = 2 * -«**£(*-<). 

s 

Thus we have for G the function 

(6.9.14) G(£,r ] ,t;T\x,y,z;t) = ^-L 

K K 



pea 

+ 2 e°l v +ri [1 - cos ^gs (t - t) ]J (sr) ds, 
Jo 



and it clearly satisfies all of the conditions prescribed above, except 
possibly the conditions at oo, which we shall presently investigate in 
some detail because of later requirements. Before doing so, however, 
we observe the important fact that G is symmetrical not only in the 
space variables f, rj, £ and x, y, z, but also in the time variables r and 
t, i.e. that 



190 WATER WAVES 

(6.9.15) G(£, rj, £; t | x, y, z; t) = G(x, y, z; t \ |, rj, £; r) and 

= G(S, rj t C;t\ x, y, z; r). 

We turn next to the discussion of the behavior of G at oo. Consider 
first the function A = 1/R — 1/R'. This function evidently will 
behave at oo like a dipole; hence if a represents distance from the 
origin it follows that A and its radial derivative A a behave as follows 
for large a: 

\A ~ l/(7 2 

[A ~ 1/(T 3 . 

On the free surface where y = we have 
A = for y = 0, 



(6.9.17) 

A y ^> 1/cr 3 for y = and large cr. 

To determine the behavior of 2?— i.e. of the integral in (6.9.14)— we 
expand [1 — cos Vgs (t — t)] in a power series in r — t and write 



(6.9.18) B = 2 



gsiv+n) j Q ( 5r ) 



g*(T-*) 2 gV(T-Q 

2! 4! 



ds. 



It is clearly legitimate to integrate term-wise for y negative. The 
formula (6.9.8) can be expressed in the form 

1 T 00 

(6.9.19) — -= e s{v+r]) J (sr) ds, 

R' Jo 

and from it we obtain 



(6.9.20) 



1 
R* 



n\P n (fi) 



R' n+1 



s n e s(y+v) J o ( 5r ) ds, 

JO 



with fi = cos 6, by a well-known formula for spherical harmonics. 
It follows, since P n (ju) are bounded functions, that the leading term 
in the asymptotic expansion of B arises from the first term in the 
square bracket. Hence the behavior of B is seen from (6.9.20) for the 
case n = 1 to be given by 

(6.9.21) B~llo\ 

for a large and any fixed values of r and t. The derivative B y is seen, 
also from (6.9.20), to behave like 1/cr 3 and the derivative B r also can 
be seen to behave like 1/cr 3 ; thus the radial derivative B a behaves in 
the same way and we have 

(6.9.22) B a — 1/cr 3 , B v ~ 1/cr 3 . 



UNSTEADY MOTIONS 191 

Summing up,- we have for the Green's function G the following behavior 

at oc : 

G ~l/(X 2 

(6.9.23) G a ~ 1/d 3 

All of these conditions hold uniformly for any fixed finite ranges in 
the values of r and t. 

We turn next to the consideration of a water wave problem of very 
general character, as follows. The space y < is filled with water and 
in addition there are immersed surfaces S t of finite dimensions having 
a prescribed motion (which, of course, must of necessity be a motion of 
small amplitude near to a rest position of equilibrium). The pressure 
on the free surface S f is prescribed for all time, and the initial position 
and velocity of the particles on the free surface and the immersed 
surfaces are given at the time t = 0. At infinity the displacement and 
velocity of all particles are assumed to be bounded. The resulting 
motion can be described for all times t > in terms of a velocity 
potential @(cc, y, z; t) which satisfies conditions of the kind studied in 
the first section of this chapter; these conditions are: 

(6.9.24) Ky,z® = ° 

in the region R consisting of the half space y < exterior to the im- 
mersed surfaces S { . On the free surface the condition 

(6.9.25) & tt +g0 y = - - p t = P(x, 0, z; t), t > 0, y = 

Q 

is prescribed, with p the given surface pressure (cf. (6.1.1 ) and (6.1.2)). 
At the equilibrium position of the immersed surfaces the condition 

(6.9.26) n = V on S if t ^ 0, 

with V the normal velocity of S i9 is prescribed. The initial position of 
Si att = is, of course, assumed known, and for the initial conditions 
otherwise we know (cf. 6.1) that it suffices to prescribe and t on 
the free surface at t = 0: 

(6.9.27) | 0(*O f *;O) =/,(*,*) 

\0 t {x,O,z;O) =f 2 (x,z). 

At oo we assume that 0, t and their first derivatives are uniformly 
bounded. 



192 WATER WAVES 

We proceed now to set up a representation for the function by 
using the Green's function obtained above. In case there are no im- 
mersed surfaces this representation furnishes an explicit solution of 
the problem, and in the other cases it leads to an integral equation for 
it. In all cases, however, a uniqueness theorem can be obtained. 

To carry out this program we begin, in the usual fashion, by applying 
Green's formula to the Green's function G and to t (rather than 0) 
in a sphere centered at the origin of radius a large enough to include 
the immersed surfaces and the singular point (f, r), f ) of the Green's 
function minus a small sphere of radius e centered at the singular point. 
Since G and t are both harmonic functions and G behaves like 1/R 
at the singular point, it follows by the usual arguments in potential 
theory that t (cc, y, z; t) is obtained in the form of a surface integral, 
as follows: 

(6.9.28) t (x, y, *;■*) = ^ if (G&tn ~ O t G n ) dS. 

e,v,c 
The symmetry of G has been used at this point. The integration varia- 
bles are |, tj, f . Even though G depends on the difference t — r the 
integral in (6.9.28) depends only on t; that is, only the singular part of 
the behavior of G matters in applying Green's formula, and the re- 
sulting expression for t depends only on the time at which t and 
tn are measured. The surface integral is taken over the boundary 
of the region just described (cf. Fig. 6.9.1), and n is the normal taken 




Fig.6.9.1. Domain for application of Green's formula 

outward from the region. The boundary is composed of three different 
parts: the portion of the sphere S a of radius a lying below the plane 
y = 0, the part S f of the plane y = cut out by the sphere S a , and 



UNSTEADY MOTIONS 



193 



the immersed surfaces S t (which might possibly cut out portions of the 
plane y = 0). 

It is important to show first of all that the contribution to the 
surface integral provided by S a tends to zero as a -> oo, and that the 
integral over S f exists as a -> oo. The second part is readily shown: 
The integrand to be studied is G0 ty — t G y . From the symmetry 
of G and (6.9.23) we see that the above integrand behaves like 1/a 2 
for large a since ty and t are assumed to be uniformly bounded at 
oo; hence the integral converges uniformly in t and r for any fixed 
ranges of these variables. To show that the integral of G0 ta — t G a 
over S a tends to zero for a -> oo requires a lengthier argument. Con- 
sider first the term t G a . Since G a behaves like 1/a 3 for large a while 
t is bounded, it is clear that the integral of this term behaves like 
1/a and hence tends to zero as a -> oo. The integral over the remaining 
term is broken up into two parts, as follows: 



(6.9.29) 



& ta GdS=\ 

Jo J (tt/2)+c5 



+ 



■ ta Ga 2 sin 6 dd d<x> 

►2jt c(nl2)+d 



r&n /• [Til 
JO Jn/2 



0<„Ga 2 sin d dd dm. 



ta 



The integrations are carried out in polar coordinates, and 6 is a small 
angle (cf. Fig. 6.9.2); the second integral represents the contribution 
from a thin strip of the sphere S a adjacent to the free surface. Since 




Fig. 6.9.2. The sphere S a 



ta is bounded and G behaves like I /a 2 for large a, it is clear that the 
absolute value of the second contribution (i.e. that from the thin strip) 
can be made less than e/2, say, if d is chosen small enough. Once d has 
been fixed, it can be seen that the contribution of the remaining part 



194 



WATER WAVES 



of S a can also be made less than e/2 in absolute value if a is taken large 
enough. If this is once shown it is then clear that the integral in ques- 
tion vanishes in the limit as a -> oo. The proof of this fact is, however, 
not difficult: we need only observe that t is by assumption bounded 
at oo and it is a well-known fact* that ta then tends to zero uniformly 
like 1/a along any ray from the origin which makes an angle ^ d with 
the plane y = 0. Thus the integrand in the first term of (6.9.29) 
behaves like 1/a and it therefore can be made arbitrarily small by 
taking a sufficiently large. Thus for t we now have the representation 

(6.9.30) t (x, y, z; t) = ~Jj(G^t, - ®fi v ) d£ dt, 

+ i- JT(G<P ( „ - <P t G„) dS, 

Si 

in which it is, of course, understood that any parts of the plane y = 
cut out by S t are omitted in the first integral. The next step is to 
integrate both sides of (6.9.30) with respect to t from to t. The result 



(6.9.31) 0(x, y, z; r) ~ 0(x, y, z; 0) 



- \\\\\G0 tv -0 t G % 
4tt JJ LJo 



dt 



d£dt + 



my- 



0fG n )dt 



dS 



v=o 



hSS 



4<7l 



(G0„ + - 0fi t ) 



o Jo 



G t + -0 tt G t )dt 



d£ dC + / 



v =o 



when G tt + gG y = for y — is used (cf. (6.9.2)) and / is intended 
as a symbol for the integral over S { . We have G = G t = for t = t; 
while for t = 0we have t = / 2 , and y \ t=0 uniquely determined hyf x ** 
from the conditions (6.9.27). In addition, we know that y J r (llg)0 tt 
= (l/g)P for t > from (6.9.25). It follows that (6.9.31 ) can be written 
in the form 



* One way to prove it is to use the Poisson integral formula expressing @ t 
at any point in terms of its values on the surface of a sphere centered at the point 
in question. Differentiation of this formula yields for any first derivative of t 
an estimate of the form Mjb where M depends only on the bound for & t on the 
sphere and b is the radius of the sphere. Finally, since our domain for d > (tt/2) + <5 
contains spheres of arbitrarily large radius at points arbitrarily far from the 
origin, the result we need follows. 

** Since 0(x, y, z; 0) is harmonic, it is uniquely determined by its boundary 
values on y = and the boundedness conditions at oo. 



UNSTEADY MOTIONS 



195 



(6.9.32; 






0(x, y, z; t 

-iff 

r?=0 

+ iff 



0(x, y, z; 0) 
1 



(G0 n +-ffi t ) 



\t=o 



Jog 



ctfcg 



G0, n <Z* 



dS 



*nj) Uo 



0,G„cfr 



dS. 



We now see that if there are no immersed surfaces S t an explicit 
solution 0(x, y, z; t) is given at once by (6.9.32) in terms of the initial 
conditions, which fix y \ t=0 and / 2 , and the condition on the free 
surface pressure fixing P — in fact, our general argument shows that 
every solution having the required properties is representable in this 
form. Consequently, the uniqueness theorem is proved for these cases. 
In particular, the Green's function constructed above is therefore 
uniquely determined since its regular part, B, satisfies the conditions 
imposed above on 0. 

In case there are immersed surfaces present the equation (6.9.32) 
does not yield the solution 0, but it does yield an integral equation 
for it in the following way (which is the standard way of obtaining 
an integral equation for a harmonic function satisfying various 
boundary conditions): One goes back to the derivation of (6.9.30), 
but considers that the singularity is at a point (x, y, z) of S t . If S t is 
sufficiently smooth (and we assume that it is) the equation (6.9.30) 
still holds, except that the factor 1/4>tz is replaced by 1/2tz, and is 
then of course given only on S { . The integration on t from to r is 
once more performed, and an equation analogous to (6.9.32) is ob- 
tained: it can be written in the form 



(6.9.33) 0{x, y, z; r) = F{x, y, z; r) — — 



2tJJ LJo 



f G„dt 



dS 



with F a known function obtained by adding together what corres- 
ponds to the first two integrals on the right hand side of (6.9.32). 
As we see, this is an integral equation for the determination of 
0(x, y, z; t) on S t . If it were once solved, the value of on S t could 
be used in (6.9.32) to furnish the values of everywhere. 

We may make use of (6.9.32) to obtain our uniqueness theorem in 
the following fashion. Suppose there were two solutions X and 2 . 
Set = ± — 2 . Then satisfies all of the conditions imposed on 
X and 2 except that the nonhomogeneous boundary conditions and 



196 WATER WAVES 

initial conditions are now replaced by homogeneous conditions, i.e. 
f 2 = P = 0; / x = and hence & y \ t=0 = since 0(x, y, z; 0) is a 
harmonic function which vanishes for y = 0, and tn = since 
n = on S t . Thus for we would have the integral representation: 

(6.9.34) 0(x, y, z; r) = - ^- IT J 0,G n <ft d9. 

Since G n behaves at oo like 1/cr 3 (cf. (6.9.23)) and values of t on the 
bounded surfaces S { are alone in question, it follows that also be- 
haves like 1/cr 3 at oo for any fixed t since the surfaces S { are bounded. 
The derivatives of could also be shown to die out at oo at least as 
rapidly as 1/cr 3 since the derivatives of G could be shown to have this 
property— for example, by proceeding in the fashion used to obtain 
(6,9.23). 

As a consequence the following function of / (essentially the energy 
integral) exists:* 

(6.9.35) E(t) = l \\([®l + ®l + <#B dxdydz + — ((0 2 t dxdz. 

R S f 

Differentiation of both sides with respect to t yields 

(6.9.36) E'(l) = jjj [(0 t ) x x + (0 t ) v (0) v + (*,),(*),] dxdydz 

R 

+ - \\0t®ttdxdz 

= f f t n dS + l [[0 t tt dxdz, 

B S f 

by application of Green's first formula, with B = Si + S f the 
boundary of R. But n = on S { and n = y = — (1 /g)0 tt on S f . 
It follows therefore that E'(t) = and £ = const. But = at 
/ = and hence E = from (6.9.35). It follows that 0,,, y , Z are 
identically zero, and thus also vanishes identically. Hence X = 2 
and our uniqueness theorem is proved. 



* It should perhaps be noted that the energy integral for the original motions 
need not, and in general will not exist, since the velocity potential and its derivati- 
ves are required only to be bounded at oo. 



SUBDIVISION C 



Waves on a Running Stream. Ship Waves 



In this concluding section of Part II made up of Chapters 7, 8, and 
9, we treat problems which involve small disturbances on a running 
stream with a free surface; that is, motions which take place in the 
neighborhood of a uniform flow, rather than in the neighborhood of 
the state of rest, as has been the case in all of the preceding chapters 
of Part II. In Chapter 7 the classical problems concerning steady 
two-dimensional motions in water of uniform (finite or infinite) depth 
are treated first. It is of considerable interest, however, to consider 
also unsteady motions (which seem to have been neglected hitherto) 
both because of their intrinsic interest and because such a study 
throws some light on various aspects of the problems concerning 
steady motions. In Chapter 8 the classical ship wave problem, in 
which the ship is idealized as a disturbance concentrated at a point 
on the surface of a running stream, is studied in considerable detail. 
In particular, a method of justifying the asymptotic treatment of the 
solution through the repeated use of the method of stationary phase is 
given, and the description of the character of the waves for both 
straight and curved courses is carried out at length. Finally, in Chapter 
9 the problem of the motion of a ship of given hull shape is treated 
under very general conditions: the ship is assumed to be a rigid body 
having six degrees of freedom and to move in the water subject only 
to the propeller thrust, gravity, and the pressure of the water, while 
the motion of the water is not restricted in any way. 



197 



CHAPTER 7 



Two-dimensional Waves on a Running Stream 
in Water of Uniform Depth 



As indicated in Fig. 7.0.1 we consider waves created in a channel 

Ay 




y= -h 

7777777777777777777777777777777777777777777777 

Fig. 7.0.1. Waves on a running stream 

of constant depth h, when the stream has uniform velocity U in the 
positive ^-direction in the undisturbed state. Such a uniform flow can 
readily be seen to fulfill the conditions derived in Chapter 1 for a 
potential flow with y = as a free surface under constant pressure. 
We assume that the motions arising from disturbances created in the 
uniform stream have a velocity potential @(x, y; t), and we set 

(7.0.1) &(x, y; t) = Ux + tp(x, y; t), — 00 < x < 00, — h < y < ?/. 

Since @(x, y; t) is a harmonic function of x and ?/ it follows that 
cp(x, y; t) is also harmonic: 



(7.0.2) 



*<P 



0. 



The function <p(x, y; t) is assumed to yield a small disturbance on the 
running stream, and we interpret this to mean that 9? and its deriva- 
tives are all small quantities and that quadratic and higher order 
terms in them can be neglected in comparison with linear terms. We 
assume also that the vertical displacement y = r)(x; t) of the free 

198 



TWO-DIMENSIONAL WAVES 199 

surface, as measured from the undisturbed position y = 0, is also a 
small quantity of the same order as cp(x, y; t). Under these circum- 
stances the dynamic free surface condition as given by Bernoulli's 
law (cf. (1.4.6)) and the kinematic free surface condition (cf.(1.4.5)) 
take the forms 



(7.0.3) * + grj + <p t + U<p x +-^-U* 
o 2 



v 



at y = 0, 



(7.0.4) rj t + Urj x - <p % 

when quadratic terms in 9? and r\ are neglected and an unessential 
additive constant is ignored in (7.0.3).* At the same time, it is proper 
and consistent in such an approximation to satisfy the free surface 
conditions at y = instead of at the displaced position y = r\. (The 
reason for this is explained in Chapter 2 — actually only for the case 
[7 = 0, but the discussion would be the same in the present case.) 
At the bottom y = — h we have the condition 

(7.0.5) cp y = at y == — h. 

In case the channel has infinite depth we replace (7.0.5) with 
(7.0.5)' cp and its derivatives up to second order are bounded at 
V = - co- 
in addition to the conditions (7.0.2) to (7.0.5) it is necessary also to 
postulate conditions at x = ± 00 and, unless the motion to be studied 
is a steady** motion with cp independent of t, it is also necessary to 
impose initial conditions at the time t = 0. The cases to be treated in 
the remainder of this chapter differ with respect to these various 
types of conditions, and we shall formulate them as they are needed. 

7.1. Steady motions in water of infinite depth with p = on the 
free surface 

If the disturbance potential op is independent of t, and if p = on 
the free surface it follows that cp(x, y) satisfies the conditions 

* It is perhaps worth noting explicitly that it would be inappropriate to 
assume that U, the velocity of the stream, is a small quantity of the same order 
as 7) and cp-. to do so would lead to the elimination of the terms in U and the 
resulting theory would not differ from that of the preceding chapters. 

** In this chapter the term "steady motion" is used in the customary way to 
describe a flow which is the same at each point in space for all times. In the 
preceding chapters we have sometimes used this term (in conformity with esta- 
blished custom in the literature dealing with wave propagation) in a different 



200 WATER WAVES 

(7.1.1) V 2 9? = 0, — oo<y^0, 

(7.1.2) <Py + — <P XX = 0, y = 0. 

g 

In addition, we require that 

(7.1.3) (p and its derivatives up to second order are bounded at oo, 

though this condition is more restrictive than is necessary. The second 
of these conditions was obtained from (7.0.3) and (7.0.4) by differen- 
tiating (7.0.3) and eliminating r\. 

It is interesting to find all functions <p(x, y) satisfying these condi- 
tions, and it is easy to do so following the same arguments as were 
used in Chapter 3.1. Using (7.1.1) we may re-write (7.1.2) in the form 

U 2 

(7.1.4) <p (p yy ■= 0, y = 0. 

S 

(This of course makes use of the fact that (p is harmonic for y = 0, 
which we assume to be true. One could easily show, in fact, that the 
free surface condition (7.1.2) permits an analytic continuation of op 
over y = 0, so that cp is actually harmonic in a domain including 
y = in its interior.) We observe that (7.1.4) is the same condition 
on q? y as was imposed on the function called cp in Chapter 3, and we 
proceed as we did there by introducing a harmonic function \p(x, y) 
through 

U 2 

(7.1.5) y) = (py -— (pyy , 2/^0. 



to 

This function vanishes on y = 0, and can therefore be continued 
analytically by reflection into the upper half plane. Since cp and its 
derivatives were assumed to be bounded in the lower half plane, it 
follows that ip is bounded in the entire plane and hence by Liouville's 
theorem it is a constant; hence ip vanishes identically since ip = 
for y = 0. Thus we have for (p y a differential equation given by (7.1.5) 
with xp = 0, and it has as its only solutions the functions 

(7.1.6) <p y = c(x)e^ V . 

Since cp y is also a harmonic function, it follows that c(x) is a solution 
of the differential equation 

d 2 c l g \ 2 
(7,1 ' 7) d? + '-' ' 



(I-.)' 



TWO-DIMENSIONAL WAVES 201 

Hence op is given by 

(7.1.8) <p(x, y) = Aeu* v cos ( S- x + a | + c x (x) 



with A and a constants and c x (x) an arbitrary function of x. By making 

d 2 c 
use of (7.1.2), however, one finds that — - = 0, and hence, that 

dx 2 

c x = const, since cp is bounded at oo. There is no loss of generality in 

taking c x = 0. The onl> solutions of our problem are therefore given 

by 



(7.1.9) <f(x, y) = AeV*"cos 



(§+•)• 



Thus the only steady motions satisfying our conditions, aside from 
a uniform flow, are periodic in x with the fixed wave length X given by 

U 2 
(7.1.10) X = 2n — . 

g 
The amplitude and phase of the motions are arbitrary. If we were to 
observe these waves from a system of coordinates moving in the in- 
direction with the constant velocity U> we would see a train of pro- 
gressing waves given by 

cp = Ae my cos m (x + Ut) 

with 

g 2n 
m = — = — . 

U 2 I 

These waves are identical with those already studied in Chapter 3 
(cf. sec. 3.2). The phase speed of these waves would of course be the 
velocity U and the wave length X would, as it should, satisfy the rela- 
tion (3.2.8) for waves having this propagation speed. In other words, 
the only waves we find are identical (when observed from a coordinate 
system moving with velocity U) with the progressing waves that are 
simple harmonic in the time and which have such a wave length that 
they would travel at velocity U in still water. 

7.2. Steady motions in water of infinite depth with a disturbing 
pressure on the free surface 

The same hypotheses are made as in the previous section, except 
that we assume the pressure on the free surface to be a function 



202 



WATElt WAVES 



p (x) =/k over the segment — a^x^a and zero otherwise, as indicat- 
ed in Fig. 7.2.1. The free surface condition, as obtained from (7.0.3) 



p = 



-a 



Pn<*> 



YY YY 



p = 



■> x 



+ a 



U 
Fig. 7.2.1. Pressure disturbance on a running stream 



and (7.0.4) by eliminating r\ and assuming r\ and q> to be independent 
of t, is now given by 



(7.2.1 



9?** + fji<Pv 



Px 

- — , on y 
Uq 



0, 



as one readily verifies. We prescribe in addition that cp and its first 
two derivatives are bounded at oo. 

The solutions (p of our problems are conveniently derived by intro- 
ducing the analytic function f(z) of the complex variable 2 = x + iy 
whose real part is (p: 

(7.2.2) /(*) = <p(x, y) + iy>(x, y). 



Since <p % 



(7.2.3) 



y) x , the condition (7.2.1) can be put in the form 



9? 3 






y> = 



v_ 



-j- const., on y = 0, 



and the constant can be taken as zero without loss of generality, since 
adding a constant to p can not affect the motion. 

We consider now only the case in which the surface pressure p is a 
constant p = p over the segment \x\ 5^ a, and zero otherwise. Since 
this surface pressure is discontinuous at x = ± a, it is necessary to 
admit a singularity at these points; we shall see that a unique solution 
of our problem is obtained if we require that 99 is bounded at these 
points while <p x and cp y behave like l/r 1-e , e > 0, with r the distance 



TWO-DIMENSIONAL WAVES 203 

from the points x = i <*> °n the free surface. (This singularity is 
weaker than the logarithmic singularity of (p appropriate at a source 
or sink.) 

In terms of f(z), the free surface condition (7.2.3) clearly can be put 
in the form 

, * /., g\, \~ — = const., \x\ < a „ 

(7.2.4) Jm lif z - 1L\ f = Uq '".- for Jm z = 0. 

I , \x\ > a 

The device of applying the boundary condition in this form seems to 
have been used first by Keldysh [K.21]. We now introduce the ana- 
lytic function F(z) defined in the lower half plane by the equation 

(7.2.5) F(z) = if z -JLf. 

This function has the following properties: 1) Its imaginary part is 
prescribed on the real axis. 2) The first derivatives of its imaginary 
part are bounded at oo, since the first two derivatives of op are assumed 
to have this property and hence f zz and f z are bounded in view of the 
Cauchy-Riemann equations. 3) Near z = ±a its imaginary part be- 
haves like l/\z ^f a| 1_£ , s > 0, as one readily sees. It is now easy to 
show that F(z) is uniquely determined,* within an additive real con- 
stant, as follows: Let G — F 1 — F 2 be the difference of two functions 
satisfying these three conditions. J> m G then vanishes on the entire 
real axis, except possibly at the points (i #, 0), and G can therefore 
be continued as a single-valued function into the whole plane except 
at the points (i a, 0). However, the singularity prescribed at the 
points (i a, 0) is weaker than that of a pole of first order, and hence 
the singularities at these points are removable. Since the first deriva- 
tives of J>m G are bounded at oo, it follows from the Cauchy-Riemann 
equations that G z is bounded at oo. Hence G z is constant, by Liouville's 
theorem, and G is the linear function: G = cz + d. Since <fm G = 
on the real axis, it follows that c and d are real constants. However, 
a term of the form cz + d on the left hand side of (7.2.5) leads to a 



term of the form olz + 8, with — — a = c, in the solution of this 

* In Chapter 4, the function F(z) given by (4.3.10) had a real part which 
satisfied identical conditions except that the condition 2) is slightly more restric- 
tive in the present case. 



204 WATER WAVES 

equation for f(z), and since f(z) is assumed to be bounded at oo. it 
follows that c = 0. 

We have here the identical situation that has been dealt with in 
sec. 3 of Chapter 4, except that it was the real part of the function 
F(z), rather than the imaginary part, that was prescribed on the real 
axis, and we can take over for our present purposes a number of the 
results obtained there. The function F(z), now known to be uniquely 
determined within an additive real constant, is given by 

(7.2.6) *■(») = -?»- log !^, 

Uqtz z + a 

which differs from F(z) as given by (4.3.12) essentially only in the 
factor i— as it should. In any case, one can readily verify that F(z) 
satisfies the conditions imposed above. We take that branch of the 
logarithm that is real for z real and \ z \ > a, and specify a branch cut 
starting at z = — a and going to oo along the positive real axis. The 
equation (7.2.5) is now an ordinary differential equation for the func- 
tion f(z) which we are seeking. 

The differential equation (7.2.5) has, of course, many solutions, 
and this means that the free surface condition and the boundedness 
conditions at oo and at the points (± a, 0) are not sufficient to ensure 
that a unique solution exists. In fact, it is clear that the non-vanishing 
solution of the homogeneous problem found in the preceding section 
could always be added to the solution of the problem formulated up 
to now. A condition at oo is needed similar to the radiation condition 
imposed in the analogous circumstances in Chapter 4. In the present 
case, the solution can be made unique by requiring that the dis- 
turbance created by the pressure over the segment \x\ ^ a should 
die out on the upstream side of the channel, i.e. at x = — oo. The only 
justification for such an assumption — aside from the fact that it 
makes the solution unique — is based on the observation that one never 
sees anything else in nature.* In sec. 7.4 we shall give a more satis- 
factory discussion of this point which is based on studying the un- 
steady flow that arises when the motion is created by a disturbance 
initiated at the time t = 0, and the steady state is obtained in the 
limit as t -> oo. In this formulation, the condition that the motion 

* Lamb [L.3], p. 399, makes use, once more, of the device of introducing 
dissipative forces of a very artificial character which then lead to a steady state 
problem with a unique solution when only boundedness conditions are prescribed 

at oo. 



TWO-DIMENSIONAL WAVES 205 

should die out on the upstream side is not imposed; instead, it turns 
out to be satisfied automatically. 

A solution of the differential equation (7.2.5) (in dimensionless 
form) has been obtained in Chapter 4 (cf. (4.3.13)) which has exactly 
the properties desired in the present case; it is: 

(7.2.7) f(z) = — — e ~u* z eu^log ^— - dt, Jm z ^ 0. 

Uqti J +ia0 t + a 

The path of integration (cf. Fig. 4.3.1) comes from ico along the posi- 
tive imaginary axis and encircles the origin in such a way as to leave 
it and the point (— a, 0) to the left. That (7.2.7) yields a solution of 
(7.2.5) is easily checked. One can also verify easily that <p = £%ef(z) 
satisfied all of the boundary and regularity conditions, except perhaps 
the condition at oo on the upstream side. In Chapter 4, however, it 
was found (cf. (4.3.15)) that f(z) behaves at oo as follows: 



(7.2.8) /(a) = 



gQ U* \zj 



for 9te z < 0, 



Thus f(z) dies out as x -> — oo, but there are in general waves of 
nonzero amplitude far downstream, i.e. at x = + oo. The uniquely 
determined harmonic function cp = tffle f(z) is now seen to satisfy all 
conditions that were imposed. 

The waves at x = -f- oo are identical (within a term of order 1/z) 
with the steady waves that we have found in the preceding section to 
be possible when the stream is subject to no disturbance (cf. (7.1.9)), 
and the wave far downstream has the wave length X = 2jzU 2 /g. 
However, we observe the curious and interesting fact (pointed out by 
Lamb [L.3], p. 404) that this wave may also vanish: clearly if 
ga/U 2 = 7i7i, n = 1, 2, . . ., <p — 0te f(z) vanishes downstream as well 
as upstream, and this occurs whenever 2ajX is an integer, i.e. whenever 
the length of the segment over which the disturbing pressure is applied 
is an integral multiple of the wave length of a steady wave in water of 
velocity U (with no disturbance anywhere). This in turn gives rise to 
the observation that there exist rigid bodies of such a shape that they 
create only a local disturbance when immersed in a running stream: 
one need only calculate the shape of the free surface— which is, of 
course, a streamline— for ga/U 2 = nn, take a rigid body having the 



206 WATER WAVES 

shape of a segment of this surface and put it into the water. (Involved 
here is, as one sees, a uniqueness theorem for problems in which the 
shape of the upper surface of the liquid, rather than the pressure, is 
prescribed over a segment, but such a theorem could be proved along the 
lines of the uniqueness proof of the analogous theorem for simple har- 
monic waves given by F. John [J.5], ) This fact has an interesting physi- 
cal consequence, i.e., that such bodies are not subject to any wave re- 
sistance (by which we mean that the resultant of the pressure forces 
on the body has no horizontal component) while in general a resistance 
would be felt. This can be seen as follows: Observe the motion from 
a coordinate system moving with velocity U in the ^-direction. All 
forces remain the same relative to this system, but the wave at -f- oo 
would now be a progressing wave simple harmonic in the time and 
having the propagation speed — U, while at — oo the wave ampli- 
tude is zero. Thus if we consider two vertical planes extending from 
the free surface down into the water, one far upstream, the other far 
downstream we know from the discussion in Chapter 3.3 that there is 
a net flow of energy into the water through these planes since energy 
streams in at the right, but no energy streams out at the left since 
the wave amplitude at the left is zero. Consequently, work must be 
done on the water by the disturbance pressure and this work is done 
at the rate RU = F, where R represents the horizontal resistance 
and F the net energy flux into the water through two planes contain- 
ing the disturbing body between them. Thus if F = — which is the 
case if the wave amplitude dies out downstream as well as upstream— 
then R = 0. This result might have practical applications. For exam- 
ple, pontoon bridges lead to motions which are approximately two- 
dimensional, and hence it might pay to shape the bottoms of the 
pontoons in such a way as to decrease the wave resistance and hence 
the required strength of the moorings. However, such a design would 
yield an optimum result, as we have seen, only at a definite velocity of 
the stream; in addition, the wave resistance is probably small com- 
pared with the resistance due to friction, etc., except in a stream 
flowing with high velocity. 

We conclude this section by giving the solution of the problem of 
determining the waves created in a stream when the disturbance is 
concentrated at a point, i.e. in the case in which the length 2a of the 
segment over which the pressure p is applied tends to zero but 
lim 2p a = P . The desired solution is obtained at once from (7.2.7); 

a-±0 

it is: 



TWO-DIMENSIONAL WAVES 207 

Pi 11 C z 1 11* 
(7.2.9) f{z) = — - e~ u* z \ _ eU J dt. 

Uqti J iao t 

This solution behaves like 1/z far upstream and like (— 2Pq/Uq) 
exp { — igz/U 2 } far downstream. Note that the amplitude downstream 
does not vanish for any special values of U in this case. It is perhaps 
also of interest to observe that f(z) behaves near the origin like i log z, 
and hence the singularity at the point of disturbance has the character 
of a vortex point; we recall that the singularity in the analogous case 
of the waves created by an oscillatory point source that were studied in 
Chapter 4 had the character of a source point, since f(z) behaved like 
log z rather than like i log z (cf. 4.3.28)), with a strength factor 
oscillatory in the time. When one thinks of the physical circumstances 
in these two different cases one sees that the present result fits the 
physical intuition. 



7.3. Steady waves in water of constant finite depth 

In water of constant finite depth the circumstances are more com- 
plicated, and in several respects more interesting, than in water of 
infinite depth. This is already indicated in the simplest case, in which 
the free surface pressure is assumed to be everywhere zero and the 
motion is assumed to be steady. In this case we seek a function cp(x, y) 
satisfying the conditions (7.0.2) to (7.0.5), with cp t and rj t both iden- 
tically zero. The boundary conditions are thus 

U 2 

(7-3.1) <p y H <Pxx = o, y = o, 

% 

and 

(7.3.2) <p y = o, y=-h. 

A harmonic function which satisfies these conditions is given by: 

(7.3.3) <p(x, y) = A cosh m(y + h) cos (mx + a) 

with A and a arbitrary constants, and m a root of the equation 

U 2 tanh mh 



(7.3.4) 



gh mh 



The condition (7.3.4) ensures that the boundary condition on the 
free surface is satisfied, as one can easily verify. It is very important 
for the discussion in this and the following section to study the roots 



208 



WATER WAVES 



of the equation (7.3.4). The curves J = tanh | and £ = (U 2 /gh) | are 
plotted in Fig. (7.3.1). The roots of (7.3.4) are of course furnished by 
the intersections £ = mh of these curves. One observes: 1) m = is 
always a root; 2 ) there are two real roots different from zero if U 2 /gh < 1 ; 



I' 


1 






/^--^S= tanh { 


I 


C = -i 





Fig. 7.3.1. Roots of the transcendental equation (XJ 2 jgh < 1) 



3) there are no real roots other than zero if U 2 /gh ^ 1; 4) if U 2 /gh = 1 
the function U 2 m — g tanh mh vanishes at m = like m 3 ; 5) since 
tan i£ = i tanh £, it follows that (7.3.4) has infinitely many pure 
imaginary roots no matter what value is assigned to U 2 /gh. 

On the basis of this discussion of the roots of (7.3.4) we therefore 
expect that no motions other than the steady flow with no surface 
disturbance (for which cp == const.) will exist unless U 2 /gh < 1. These 
waves are then seen to have the wave length appropriate for simple 
harmonic waves of propagation speed c = U in water of depth h, as 
can be seen from (3.2.1), (3.2.2), and (3.2.3). It is possible to give a 
rigorous proof of this uniqueness theorem — which holds when no con- 
ditions at oo other than boundedness conditions are imposed — by 
making use of an appropriate Green's function, or by making use of 
the method devised by Weinstein [W.7] for simple harmonic waves in 
water of finite depth, but we will not do so here. 

More interesting problems arise when we suppose that steady waves 
are created by disturbances on the free surface, or perhaps also on the 
bottom. Mathematically this means that a nonhomogeneous boundary 
condition would replace one, or perhaps both, of the homogeneous 



TWO-DIMENSIONAL WAVES 209 

boundary conditions (7.3.1) and (7.3.2). In addition, as we infer from 
the discussion of the preceding section, it is also necessary in general 
to prescribe a condition of "radiation" type at oo in addition to boun- 
dedness conditions, and an appropriate such condition is that the 
disturbance should die out upstream. In the present problem, how- 
ever, the additional parameter furnished by the depth of the water 
leads to some peculiarities that are conditioned in part by the differ- 
ence in behavior of the solutions of the homogeneous problem in their 
dependence on the parameter U 2 /gh: Since the only solution of the 
homogeneous problem in the case U 2 jgh ^1 is <p = 0, one expects 
that the solution of the nonhomogeneous problem will be uniquely 
determined in this case without the necessity of prescribing a radiation 
condition at oo. However, if U 2 jgh < 1 it is clear that the nonhomo- 
geneous problem can not have a unique solution unless a condition — 
such as that requiring the disturbance to die out upstream — is 
imposed that will rule out the otherwise possible addition of the non- 
vanishing solution of the homogeneous problem. These cases have been 
worked out (cf. Lamb [L.3], p. 407) with the expected results, as 
outlined above, for U 2 !gh > 1 and U 2 /gh < 1, but the known re- 
presentations of these solutions for the steady state make the wave 
amplitudes large for U 2 /gh = 1 and \x\ large. 

We shall not solve these steady state problems directly here be- 
cause the peculiarities — not to say obscurities — indicated above can 
all be clarified and understood by re-casting the formulation of the 
problem in a way that has already been employed in the previous 
chapter (cf. sec. 6.7)). The basic idea (cf. Stoker [S.22]) is to abandon 
the formulation of the problem in terms of a steady motion in favor of 
a formulation involving appropriate initial conditions at the time 
t = 0, and afterwards to make a passage to the limit in the solutions 
for the unsteady motion by allowing the time to tend to oo. As was 
indicated in sec. 6.7, the advantage of such a procedure is that the 
initial value problem, being the natural dynamical problem in New- 
tonian mechanics (while the steady state is an artificial problem), has 
a unique solution when no conditions other than boundedness con- 
ditions are imposed at oo. If a steady state exists at all, it should then 
result upon letting t -> oo, and the limit state would then automati- 
cally have those properties at oo which satisfy what one calls radia- 
tion conditions, and which one has to guess at if the steady state 
problem is taken as the starting point of the investigation. 

We shall proceed along these lines in the next section in attacking 



210 WATER WAVES 

the problem of the waves created in a stream of uniform depth when 
a disturbance is created in the undisturbed uniform stream at the 
time t = 0. The subsequent unsteady motion will be determined when 
only boundedness conditions are imposed at oo. It will then be seen 
that the behavior of the solutions as t -> oo is indeed as indicated 
above, i.e. the waves die out at infinity both upstream and down- 
stream when U 2 /gh > 1, that they die out upstream but not down- 
stream when U 2 /gh < 1. One might be inclined to say: "Well, what of 
it, since one guessed the correct condition on the upstream side any- 
way?" However, we now get a further insight, which we did not 
possess before, i.e. that for U 2 /gh = 1 there just simply is no steady 
state when t -> oo although a uniquely determined unsteady motion 
exists for every given value of the time t. In fact it will be shown that 
the disturbance potential becomes infinite like t 2/3 at all points of the 
fluid when t -> oo and U 2 /gh = 1, and that the velocity also becomes 
infinite everywhere when t -> oo. 

7.4. Unsteady waves created by a disturbance on the surface of a 
running stream 

The boundary conditions on the disturbance potential <p(x, y; t) at 
the free surface (cf. Fig. 7.0.1 and equations (7.0.3) and (7.0.4)) are 

V U 2 

(7.4.1) il + grj + <p t + Ucp x + — = 0, 

Q 2 

(7.4.2) rj t + Urj x - <p p = 0, 

to be satisfied at y = for all times t > 0. The quantity p = p(x; t) 
is the pressure prescribed on the free surface. At the bottom y — — h 
we have, of course, the condition 

(7.4.3) <p y = 0, t^0. 

At the initial instant i = 0we suppose the flow to be the undisturbed 
uniform flow, and hence we prescribe the initial conditions: 

(7.4.4) (p(x, y; 0) = rj(x; 0) = p(x; 0) = 0. 

From (7.4.1), which we assume to hold at t = 0, we thus have the 
condition 

(7.4.5) <p t (x, y; 0) = 0. 

Finally, we prescribe the surface pressure p for t > 0: 

(7.4.6) p = p(x), t > 0. 



TWO-DIMENSIONAL WAVES 211 

(The surface pressure is thus constant in time.) At oo we make no 
assumptions other than boundedness assumptions. We shall not 
formulate these boundedness conditions explicitly: instead, they are 
used implicitly in what follows because of the fact that Fourier trans- 
forms in x for — oo < x < oo are applied to cp and p and their 
derivatives. Of course, this means that these quantities must not only 
be bounded but also must tend to zero at 00, and this seems reasonable 
since the initial conditions leave the water undisturbed at 00. 

We have, therefore, the problem of finding the surface elevation 
rj(x; t) and the velocity potential (p(x, y; t) in the strip — h ^ y f^ 0, 
— 00 < x < 00, which satisfy the conditions (7.4.1) to (7.4.6). We 
begin the solution of our problem by eliminating the surface elevation 
tj from the first two boundary conditions to obtain: 

(7.4.7) cp u + U*(p xx + 2U<p xt +g(p y = - — p x , at y = 0. 

Q 

The Fourier transform with respect to x is now applied to (p xx -\-<p yy = 
to yield (cf. sec. 6.3): 

(7.4.8) v vy - s*? = 0, 

where the bar over cp refers to the transform cp = cp(s, y; t) of 99. From 
(7.4.3) we have cp y = for y = — h; hence (p, in view of (7.4.8) must 
be of the form 

(7.4.9) qj(s, y; t) = A(s; t) cosh s(y + h), 

with A(s; t) a function to be determined. The transform is next applied 
to (7.4.7) with the result: 

- - - isU - 

(7.4.10) <p tt + 2isU(p t + g<p y - U 2 s 2 (p = p, at y = 0, 

9 

and this yields, from (7.4.9) for y = 0, the differential equation 

isUp 



(7.4.11) A u + 2isUA t + [gs tanh sh - s 2 U 2 ]A 



q cosh sh 



Here p(s) is of course the transform of p(x). As initial conditions at 
t = 0forA(s; t)we have from (7.4.4) and (7.4.5) the conditions (again 
in conjunction with (7.4.9)): 

(7.4.12) A(s; 0) = A t (s; 0) = 0. 

The function A(s; t) is then easily found; it is 



212 



WATER WAVES 



(7.4.13) A(s;t) 



isUp 



g cosh sh 



s 2 U 2 — gs tanh sh 



-it (sll + \/gs tanh sh) 



2 Vgs tanh sh sU + Vgs tanh sA 

"1 p~ it [sU — V gs tanhsft) 



2 V gs tanh sh sU — Vgs tanh sh 

The solution 99(0?, y; t) of our problem is of course now obtained by 
inverting cp(s, y; t): 



(7.4.14) <p(x,y;t) 



V%n 



A(s; t) cosh s(y + h) e isx ds. 



The path of integration is the real axis. One finds easily that the 
integrand behaves for large s like e^ y /s, since the denominators of 
the terms in the square brackets in (7.4.13) behave like s 2 , the ratio 
cosh s(y -f- h)/eosh sh behaves like e^ y for large s, and p(s) tends to 
zero at 00 in general. Since y is negative (cf. Fig. 7.0.1) it is clear that 
the integral converges uniformly. (We omit a discussion of the be- 
havior on the free surface corresponding to y — 0, although such a 
discussion would not present any real difficulties.) Upon examining 
the function A(s; t) in (7.4.13) it might seem that it has singularities 
at zeros of the denominators (and such zeros can occur, as we shall see) 
but in reality one can easily verify that the function has no singulari- 
ties when the three terms in the square brackets are taken together — 
or, as one might also put it, any singularities in the individual terms 
cancel each other. Thus the solution given by (7.4.14) is a regular 
harmonic function in the strip — h ^ y < for all time t, or, in 
other words, a motion exists no matter what values are given to the 
parameters. In addition, the fact that the integral exists ensures that 
(p (and also its derivatives) tends to zero for any given time when 
\x\ -> 00 — this is the content of the so-called Riemann-Lebesgue 
theorem. This means that the amplitude of the disturbance dies out 
at infinity at any given time t — a not unexpected result since a certain 
time must elapse before any appreciable effects of a disturbance are 
felt at a distance from the seat of the disturbance.* 

However, we know from our earlier discussion (and from everyday 

* It should be pointed out once more that disturbances propagate at infinite 
speed since our medium is incompressible. Each Fourier component, however, 
propagates with a finite speed. 



TWO-DIMENSIONAL WAVES 213 

observation of streams, for that matter) that as t -> oo it may happen 
that a disturbance also propagates downstream as a wave with non- 
vanishing amplitude. Our main interest here is to study such a passage 
to the limit. It is clear that one cannot accomplish such a purpose 
simply by letting t -> oo in (7.4.14), since, for one thing, the transform 
<p of <p cannot exist if cp does not tend to zero at oo. What we wish to do 
is to consider the contributions of the separate items in the brackets 
in (7.4.13), and to avoid any singularities caused by zeros in their 
denominators by regarding A (s; t) as an analytic function in the neigh- 
borhood of the real axis of a complex s-plane and deforming the path 
of integration in (7.4.14) by Cauchy's integral theorem in such a way 
as to avoid such singularities. One can then study the limit situation 
as t -> oo. 

In carrying out this program it is essential to study the separate 
terms defining the function A(s; t) given by (7.4.13). To begin with, 
we observe that the function V gs tanh sh can be defined as an analytic 
and single-valued function in a neighborhood of the real axis since 
the function s tanh sh has a power series development at s = that 
is valid for all s and begins with a term in s 2 , and, in addition, the 
function has no real zero except ,9 = 0. Once the function Vgs tanh sh 
has been so defined, it follows that each of the terms in (7.4.13) is an 
analytic function in a strip containing the real axis except at real zeros 
of the denominators. It is important to take account of these zeros, 
as we have already done in sec. 7.3. For our present purposes it is 
useful to consider the function 

/U 2 \ 

(7.4.15) Wis) = gs — .sh — tanh sh = s 2 U 2 — gs tanh sh 



= (sU + Vgs tanh sh)(sU — Vgs tanh sh) 

With reference to Fig. 7.3.1 above and the accompanying discussion, 
one sees that there are at most three real zeros of the function W(s): 
s = is in all cases a root, and there exist in addition two other real 
roots if the dimensionless parameter gh/U 2 is greater than unity. Also, 
it is clear that if gh/U 2 =£ 1 the origin is a double root of W(s), but is 
a quadruple root if gh/U 2 — 1. In case gh/U 2 > 1 the real roots i /3 
of W(s) are simple roots. (It might be noted in passing that W(s) has 
infinitely many pure imaginary zeros ±i/? n , n =■ 1,2, . . ..) 

It follows at once that if we deform the path of integration in 



214 



WATER WAVES 



(7.4.14) from the real axis to the path P shown in Figure 7.4.1 we can 
consider separately the contributions to the integral furnished by each 
of the three items in the square brackets in (7.4.13), since the separate 



p 



X7 



+$ 



^7 



V 



Fig. 7.4.1. The path P in the s-plane 

integrals would then exist. This we proceed to do, except that we pre- 
fer to consider the velocity components <p x and q? y of the disturbance 
rather than (p itself. For <p x we write* 



(7.4.16) 



<Pa 



<P { X S) + 9 



(t) 



with <p£ and q>® defined (in accordance with (7.4.13) and (7.4.14)) as 
follows: 



(7.4.17) (pi 



U C p(s)s 2 cosh s(y + h) 



In J 



7.4.18) (p 



(*) 



■qV 2 ™ J 



q\/2tz J p W(s) cosh sh 
p(s)s 2 cosh s(y + h) 



ds, 



2q^/2ti J P cosh sh \/gs tanh sh 



itfls) 



/+(«) 



itf (s) 



/-(») 



ds. 



The functions W(s), /_(«), and f + (s) have been defined in (7.4.15). 
Evidently the notation cp x s \ (p® has been chosen to point to the fact 
that <?9 (s) should yield the steady part of the motion while cp^ should 
furnish "transients" which die out as t -> oo. This is indeed the case, 
as we now show, at least when the parameter gh/U 2 is not equal to 
unity, its critical value. 

Consider first the case gh/U 2 < 1. In this case there are no singu- 
larities on the real axis, even at the origin (cf. (7.4.18)), since / + and 
/_ vanish to the first power and y/gs tanh sh vanishes to the first 
power also at s = 0. Since p(s) is regular at s = and s 2 occurs in the 
numerator of the integrand our statement follows. Consequently the 
path P can be deformed back again into the real axis. In this case the 



* The discussion would differ in no essential way for y y instead of cp x . 



TWO-DIMENSIONAL WAVES 215 

behavior of <p® for large t can be obtained by the principle of station- 
ary phase (cf. sec. 6.8). In the present case the functions f + (s) and 
f_(s) have non-vanishing first derivatives for all s, and consequently 
cp {t) -> at least like 1/t since there are no points where the phase is 
stationary. (Here and in what follows no attempt is made to give the 
asymptotic behavior with any more precision than is necessary for 
the purposes in view. ) As t -> oo therefore we obtain the steady state 
solution <p { jK The behavior of (p { f for \x\ large is also obtained at 
once: one sees that the integrand in (7.4.17) has no singularities in 
this case also, and it follows at once from the Riemann-Lebesgue 
theorem that <p (s) -> as \x\ -> oo. Thus a steady state exists, and 
it has the property that the disturbances die out both upstream and 
downstream. 

We turn next to the more complicated case in which gh/U 2 > 1. 
The integrand for cp w has no singularity at the origin, but it has 
simple poles at s = ± f$ (cf. Figure 7.4.1) furnished by simple zeros 
of f_(s) at these points. Again we show that <p® -^ as t -> oo. 
Consider first the contribution of the semicircles at s = i /?. (Since 
s = is not a singularity, we deform the path back into the real axis 
there.) In the lower half-plane near s = ± (3 one sees readily that 
f_{s) has a negative imaginary part, and thus the exponent in 
exp {— itf_(s)} has a negative real part, since f_(s) is real on the real 
axis and its first derivative f_(s) is positive there (so that /_(<?) be- 
haves like c(s T/?) with c a positive constant). Thus for any closed 
portion of the semicircles which excludes the end-points the contribu- 
tion to the integral tends to zero as t -» oo, and hence also for the 
whole of the semicircles. On the straight parts of the path the prin- 
ciple of stationary phase can be used again to show that <p$ -» as 
t -> oo. In fact, this function behaves like ljy/t since one can easily 
verify that f_(s) has exactly two points of stationary phase, i.e. two 
points ± £ where f'_(± fi ) = and /"(± O ) ^ 0. (The point s = /9 
lies between the origin and the point s = (3 where f_(s) vanishes.) 
Thus the steady state is again given by <p (s) . However, unlike the pre- 
ceding case, the steady state does not furnish a motion which dies 
out both upstream and downstream. This can be seen as follows. 
Consider first the behavior upstream, i.e. for x < 0. On the semicircu- 
lar parts of the path P in the lower half-plane we see that the expo- 
nent in e isx in (7.4.17) has a negative real part, and therefore by the 
same argument as above, these parts of P make contributions which 
vanish as x -> — oo. The straight parts of P also make contributions 



216 WATER WAVES 

which vanish for large x (either positive or negative), by the Riemann- 
Lebesgue theorem. Thus the disturbance vanishes upstream. On the 
downstream side, i.e. for x > 0, we cannot conclude that the semi- 
circular parts of P make vanishing contributions for large x since the 
exponent in e lsx now has a positive real part. We therefore make use 
of the standard procedure of deforming the path P through the poles 
at s = ± P and subtracting the residues at these poles. It is clear that 
the semicircles in the upper half-plane yield vanishing contributions 
to 99(f) when x -> + oo: the argument is the same as was used above. 
This leads to the following asymptotic representation (obtained from 
the contributions at the poles), valid for x large and positive: 

q cosh phW (p) 

Here W'(fl) ^ is the value of the derivative of W (cf. (7.4.15)) at 
s = ft, and the fact that W(f$) is an odd function has been used. In 
particular, if the surface pressure p(x) were given by the delta func- 
tion p(x) = d(x), i.e. if the disturbance were caused by a concentrated 
pressure point at the origin, (7.4.19) would yield 

2B 2 U cosh 8(y -f h) . n 

7.4.19 ), wjx, y; oo ~ J- HK& } sin Bx 

11 rxX U ' Q cosh phW'(P) H 

since the transform of d(x) is 1/\/2tz. Another interesting special 
case is that in which p(x) is a constant p over the interval — a 5^ x 
5^ a and zero over the rest of the free surface. In this case p = 
(2p /y / 2ji:)(sm sa)js and cp x behaves for large positive x and t as follows: 

4p 6U cosh B(y + h) . a . Q 

(7.4.19) 2 cp x (x, y; co) - -^ u ™ ta . sin P" sm fa- 

q cosh phW (p) 

This yields the curious result (mentioned above) that under the pro- 
per circumstances the disturbance may die out downstream as well as 
upstream; it will in fact do so if pa = ?iji, i.e. if the length 2a of the 
segment over which the disturbing pressure is applied is an integral 
multiple of the wave length at oo— which is, in turn, fixed by the 
velocity U and the depth h. 

Finally we consider the critical case gh/U 2 = 1, and begin by dis- 
cussing the behavior of the time dependent terms in cp as t -> oo. For 
this purpose it is convenient to deal first with the time derivative of 
this function: 



TWO-DIMENSIONAL WAVES 217 



u 

(7.4.20) <p 



2q^/2jc 



P cosh sh\/gs tanh sh 






The integrand has no singularities on the real axis and consequently 
the path P can be deformed into the real axis. Thus the principle of 
stationary phase can be employed once more. Since the derivative of 
f + (s) = sU + \/gs tanh sh evidently does not vanish for any real s 
while the derivative of /_(<?) has one zero at s = 0, it follows that the 
leading term in the asymptotic development of cp^ for large t arises 
from the term exp {— itf_(s)}. Since, in addition, /"(0) = but /"'(0) 
^0 we have (cf. sec. 6.8): 

(7.4.21) w®~Ap(0).—- — , A = const. ^ 0. 

Since p(Q) is in general different from zero, it follows that cp'f behaves 
like r 1/3 and hence that <p {t) becomes infinite everywhere (for all x 
and y, that is) like t v% as t -> oo.* Thus a steady state does not exist 
if one considers it to be the limit as t -> oo. It might be thought that 
the existence in practice of dissipative forces could lead to the vanish- 
ing of the transients and thus still leave the steady state cp ^ as given 
by (7.4.17) as a representation of the final motion. That is, however, 
also not satisfactory since 9? (s) becomes unbounded for x large when 
gh/U 2 = 1: at the origin there is a pole of order two since W(s) be- 
haves like s 4 and consequently the term isx in the power series for e lsx 
leads to a contribution from this pole which is linear in x. In linear 
theories based on assuming small disturbances one is reconciled to 
singularities and infinities at isolated points, but hardly to arbitrarily 
large disturbances in whole regions. All of this suggests that the 
reasonable attitude to take in these circumstances is that the linear 
theory, which assumes small disturbances, fails altogether for flows 
at the critical speed U 2 /gh = 1 and that one should go over to a non- 



* It might seem odd that we have chosen to discuss the function tpjf' rather 
than the function q>^ (as we did in the other cases). The reason is that the asymp- 
totic behavior of qffl is not easily obtained directly by the method of stationary 
phase in the present case since the coefficient of the leading term in this develop- 
ment would be zero. However, one could show (by using Watson's lemma, for 
example, which yields the complete asymptotic expansion of the integral) that 
(p^ behaves like t~ 2//s , and hence that w^ behaves like t /z . 

*% ' X 



218 WATER WAVES 

linear theory in order to obtain reasonable results from the physical 
point of view. In Chapter 10.9, which deals with the solitary wave (an 
essentially nonlinear phenomenon), we shall see that such a steady 
wave exists for flows with velocities in the neighborhood of the 
critical value. 



CHAPTER 8 

Waves Caused by a Moving Pressure Point. Kelvin's 
Theory of the Wave Pattern Created by a Moving Ship 

8.1. An idealized version of the ship wave problem. Treatment by the 
method of stationary phase 

The peculiar pattern of the waves created by objects moving over 
the surface of the water on a straight course has been noticed by 
everyone: the disturbance follows the moving object unchanged in 
form and it is confined to a region behind the object that has the same 
v-shape whether the moving object is a duck or a battleship. An ex- 
planation and treatment of the phenomenon was first given by 
Kelvin [K.ll], and this work deserves high rank among the many 
imaginative things created by him. As was mentioned earlier, Kelvin 
invented his method of stationary phase as a tool for approximating 
the solution of this particular problem, and it is indeed a beautiful 
and strikingly successful example of its usefulness. 

It should be stated at once that there is no notion in this and the 
next following section of solving the problem of the waves created by 
an actual ship in the sense that the shape of the ship's hull is to be 
taken into account; such problems will be considered in the next 
chapter. For practical purposes an analysis of the waves in such cases 
is very much desired, since a fraction— even a large fraction if the 
speed of the ship is large— of the resistance to the forward motion of a 
ship is due to the energy used up in maintaining the system of gravity 
waves which accompanies it. The problem has of course been studied, 
in particular, in a long series of notable papers by Havelock,* but the 
difficulties in carrying out the discussion in terms of parameters which 
fix the shape of the ship are very great. Indeed, a more or less com- 
plete discussion of the solution to all orders of approximation even in 
the very much idealized case to be studied in the present chapter, is 
by no means an easy task— in fact, such a complete discussion, along 

* References to some of these papers will be given in the next chapter. 

219 



220 WATER WAVES 

lines quite different from those of Kelvin, has been carried out only 
rather recently by A. S. Peters [P. 4] (cf. also the earlier paper by 
Hogner [H.16]). However, we shall follow Kelvin's procedure here in 
a general way, but with many differences in detail. 

The problem we have in mind to discuss as a primitive substitute 
for the case of an actual ship is the problem of the surface waves 
created by a point impulse which moves over the surface of the water 
(assumed to be infinite in depth). We shall take the solution of section 
6.5 for the wave motion due to a point impulse and integrate it along 
the course of the "ship"— in effect, the surface waves caused by the 
ship are considered to be the cumulative result of impulses delivered 
at each point along its course. The result will be an integral represen- 
tation for the solution, in the form of a triple integral, which can be 
discussed by the method of stationary phase. However, it is necessary 
to apply the method of stationary phase three times in succession, and 
if this is not done with some care it is not clear that the approximation 
is valid at all; or what is perhaps equally bad from the physical point 
of view, it may not be clear where the approximation can be expected 
to be accurate. Thus it seems worth while to consider the problem with 
some attention to the mathematical details; this will be done in the 
present section, and the interpretation of the results of the approxima- 
tion will be carried out in the next section (which, it should be said, 
can be read pretty much independently of the present section). 

From section 6.4 the vertical displacement* r\(x, y, z; t) of the water 
particles due to a point impulse applied on the surface at the origin 
and at the time t = can be put in the form 

1 /» oo /• re/2 

(8.1.1) r)(x,y,z;t)= — a sin ot- e mv mdm \ cos (?nr cos j3) d (3 

2ngeJ-o Jo 

in which a 2 = gm and r 2 = x 2 + z 2 . We have replaced the Bessel 
function J (mr) by its integral representation 

2 r n i 2 

J (mr) = — cos (mr cos [5) dfi 

for reasons which will become clear in a moment. As we have 
indicated, our intention is to sum up the effect of such impulses 
as the "ship" moves along its course C. The notations to be used for 

* Actually, we have considered only the displacement of the free surface in 
that section, but it is readily seen that (8.1.1) furnishes the vertical displacement 
of any points in the water. 



WAVE PATTERN CREATED BY A MOVING SHIP 



221 



this purpose are indicated in Figure 8.1.1, which is to be considered as 
a vertical projection of the free surface on any plane y = const. The 
course of the ship is given in terms of a parameter t by the relations 

L = «l(0 



(8.1.2) \' x ~ ±x ~' O^t^T, 

l*i = z i(0 

and t is assumed to mean the time required for the ship to travel 
from any point Q(x v %) on its course to its present position at the 




P(x,z) 

Fig. 8.1.1. Notation for the ship wave problem 

origin. We seek the displacement of the water at (x, y, z) when the 
ship is at the origin; it is therefore determined by the integral 

(8.1.3) rj(x,y,z) 

= — k(t) dt \ a sin at e my m dm cos (mr cos ($)d[$ 

ZngQJo Jo Jo 

In this formula k(t) represents the strength of the impulse, which we 
might reasonably assume to be constant if the speed of the ship is 
constant; this constant is therefore the only parameter at our disposal 
which might serve to represent the effect of the volume, shape, etc. 
of a ship's hull. We write the last relation in the form 

Tcoti/2 

(8.1.4) r](x,y,z)=K ((( G me m y[e i{ot - mrcos ^ + e i{at+mr cos ^] dp dm dt 

000 

with the understanding that the imaginary part of the integral is to be 
taken. (K is a constant the value of which is not important for the 



222 WATER WAVES 

discussion to follow.) It should be noted that r 2 = (x — x x ) 2 + 
(z — z x ) 2 . Since y < 0, the integral converges strongly because of the 
exponential factor. 

One of the puzzling features (to the author, at least) of existing 
treatments of the problem by the method of stationary phase is that 
it is not made clear what parameter is large in the exponentials as the 
method is applied to each of the three integrals in turn, so that one is 
not quite sure whether there might not be an inconsistency. The 
matter is easily clarified by introduction of appropriate dimensionless 
quantities, as follows (cf. Figure 8.1.1): 

(8.1.5) 

' x = R cos a, x x = R x cos a x , z = R sin a, z x = R x sin a l5 



r = R VfAcosa! — cos a) 2 + (Asina! — sin a) 2 = R . I, 

T = ct/R, RJR = l K = g —, m = ^-i 2 . 

4>c 2 4<r 2 

Here the quantity c represents the speed of the ship in its course. It 
should be noted that x, y, and z are held fixed— they represent the 
point at which the displacement is to be observed — , but that x l9 z x 
(and hence R x and a x ), and r all depend on t. We have also introduced 
a new variable of integration £, replacing m, which depends on t. The 
Jacobian d(m, t)/d(i;, r) has the value gt 2 R£/(2cr 2 ) and hence vanishes 
only for t = 0. In terms of the new quantities the integral (8.1.4) is 
found to take the form: 

(8.1.6) rj{x, y, z) 

rV T v3 T 5^4 xtH 2 v ( . (2|-s £2 cos/3)t 2 . (2|+| 2 cos ft t 2 1 

= 4K III ^± e «rj e " i +e" > 1 Idpdfdx, 



where t = cT/R. 

Again we remark that the integral converges uniformly for y < 0. 
However, the integrand has a singularity if the point (x, y, z) happens 
to be vertically under a point on the course of the ship: in such a case 
we have R = R x (i.e. A = 1 ), and a = a l5 so that I = for a certain 
value t ^0 in the interval ^ r ^ t . Because of the exponential 
factor, the integral continues to exist, however. Indeed, one sees 
from (8.1.4) that taking r = does not make the integrand singular; 



WAVE PATTERN CREATED BY A MOVING SHIP 223 

the fact that a singularity crops up in (8.1.6) arises from our choice 
of the variable £ which replaces m. This disadvantage caused by intro- 
duction of the new variables is much more than outweighed by the 
fact that we now can see that the approximation by the method of 
stationary phase depends only on one parameter, i.e. the parameter 
x = gR/4iC 2 in the exponentials. We can expect the use of the method 
of stationary phase to yield an accurate result if this parameter is 
large, and that in turn is certainly the case if R is large, i.e. for points 
not too near the vertical axis through the present location of the ship. 

The application of the method of stationary phase to the integral 
in (8.1.6) can now be justified by an appeal to the arguments used in 
section 6.8. In doing so, the multiple integral is evaluated by inte- 
grating with respect to each variable in turn; at the same time, the 
integrands are replaced by their asymptotic representations as fur- 
nished by the method of stationary phase. One need only observe, in 
verifying the correctness of such a procedure, that the integrands 
remain, after each integration, in a form such that the arguments of 
that section apply— in particular that they remain analytic functions 
of their arguments provided only that points (x, y, z) on or under the 
ship's course are avoided* — and that an asymptotic series can be in- 
tegrated termwise. It is not difficult to see that the contributions to 
rj(x, y, z) of lowest order in \jx are made by arbitrarily small domains 
containing in their interiors a point where the derivatives cpp, <p§, <p r of 
the phase (p = (2£ — £ 2 cos /?)t 2 /Z(t) vanish simultaneously. 

Even for points on the ship's course the argument of section 6.8 
will still hold provided that no stationary point of the phase y occurs 
for a value of x such that Z(t) = 0: the reason for this is that the 
assumption of analyticity was used in section 6.8 only to treat a 
neighborhood of a point of stationary phase, while for other segments 
of the field of integration only the assumptions of integrability and 
the possibility of integration by parts are needed. It happens that the 
cases to be treated later on are such that l(r) does not vanish at any 
points of stationary phase, and hence for them the asymptotic 
approximation is valid also for points on the ship's course. 

There is one further mathematical point to be mentioned. The 

rb 
* In section 6.8 the integrals studied were of the form rp(x) exp {ik(p{x)} dx, 

r b J a 

while here the integral is of the form xp(x, k) exp {ik<p(x)} dx. However, one 

can verify that the argument used in section 6.8 can easily be generalized to 
include the present case. 



224 WATER WAVES 

above discussion requires that we take y < 0, and it is not entirely 
clear that the passage to the limit y -» is legitimate in the approxi- 
mate formulas, so that the validity of the discussion might be thought 
open to question for points on the free surface. Indeed, it would appear 
to be difficult to justify such a limit procedure for the integral in 
(8.1.1), for instance, since it certainly does not converge if we set 
y = since the integrand then does not even approach zero as m -> oo. 
However, this is a consequence of dealing with a point impulse. If we 
had assumed as model for our ship a moving circular disk of radius a 
over which a constant distribution of impulse is taken, the result for 
the vertical displacement due to such a distributed impulse applied 
at t = could be shown to be given by 

rj(x, y, z; t) = K ± \ a sin at • e my J (mr)J 1 (ma) dm 

with J^ma) the Bessel function of order one and K 1 a certain constant. 
This integral converges uniformly for y ^ 0, as one can see from the 
asymptotic behavior of J (mr) and J 1 (ma). Consequently rj(x, y, z; t) 
is continuous for y — 0. On the other hand, if the radius a of the disk 
is small the result cannot be much different from that for the point 
impulse. Thus we might think of the results obtained in the next 
section, which start with the formula (8.1.1 ) for a point impulse, as an 
approximation on the free surface to the case of an impulse distributed 
over a disk of small radius. 

It has already been mentioned that the problem under discussion 
here has been treated by A. S. Peters [P. 4] by a different method. 
Peters obtains a representation for the solution based on contour 
integrals in the complex plane, which can then be treated by the 
saddle point method to obtain the complete asymptotic development 
of the solution with respect to the parameter x defined above, while 
we obtain here only the term of lowest order in such a development. 
However, the methods used by Peters lead to rather intricate deve- 
lopments. 

8.2. The classical ship wave problem. Details of the solution 

In the preceding section we have justified the repeated application 
of the method of stationary phase to obtain an approximate solution 
for the problem of the waves created when a point impulse moves over 
the surface of water of infinite depth. In particular, it was seen that 
the approximation obtained in that way is valid at all points on the 



WAVE PATTERN CREATED BY A MOVING SHIP 



225 



surface of the water not too near to the position of the "ship" at the 
instant when the motion is to be determined (provided only that a 
certain condition is satisfied at points on the ship's course). In this 
section we carry out the calculations and discuss the results, returning 
however to the original variables since no gain in simplicity would be 
achieved from the use of the dimensionless variables of the preceding 
section. 

Kelvin carried out his solution of the ship wave problem for the 
case of a straight line course traversed at constant speed. Up to a 
certain point there is no difficulty in considering more general courses 




Fig. 8.2.1. Notation for the ship wave problem 

for the ship. In Figure 8.2.1 we indicate the course C as any curve 
given in terms of a parameter t by the equations 

i = x x (t) 



(8.2.1) \ "* " AV ' for ^ t ^ T. 

[ *i = *i(0 

The parameter t is taken to represent the time required for the ship 
to pass from any point (zc v z ± ) to its present position at the origin 0, 
but it is convenient to take t = to correspond to the origin so that 
the point (x v y x ) moves backward along the ship's course as t increases. 
The shape of the waves on the free surface is to be determined at the 
moment when the ship is at the origin. The #-axis is taken along the 
tangent to the course C, but is taken positive in the direction opposite 
to the direction of travel of the ship, Since we have taken t = at the 
origin the parameter t in (8.2.1) is really the negative of the time; as 
a consequence the tangent vector t to C at a point Q(x v y x ) as given by 



(8.2.2) t = P, ^ 

\dt dtj 

is in the direction opposite to that of the ship in traversing the course 



226 WATER WAVES 

C. The speed c(t) of the ship is the length of the vector t and is given by 
(8.2.2)! c(t) 



m + m 



The point P(x, z) is the point at which the amplitude of the surface 
waves is to be computed; it is located by means of the vector r: 

(8.2.3) r = (x — x v z — z x ). 

The angle 6 indicated on the figure is the angle (^ n) between the 
vectors r and — t. 

As we have stated earlier, the surface elevation rj(cc, z) at P{x, z) 
is to be determined by integrating the elevations due to a point im- 
pulse moving along C. The effect of an impulse at the point Q is ass- 
sumed to be given by the approximate formula (6.5.15), in which, 
however, we omit a constant multiplier which is unessential for the 
discussion to follow: i 

— t 3 2t 2 

(8.2.4) rj(r; t) ~ sin — . 

r 4 4r 

In other words, we assume that the formula (8.1.1) for the surface 
elevation r\ has been approximated by two successive applications 
of the method of stationary phase. This formula yields the effect 
at time t and at a point distant r from the point where the impulse 
was applied at the time t = 0; it therefore applies in the present situa- 
tion with 

(8.2.5) r 2 = {x - x x f + (z - zj 2 , 

since t does indeed represent the length of time elapsed since the 
"ship" passed the point Q on its way to its present position at 0. The 
integrated effect of all the point impulses is therefore given by 



C T l 3 at 2 
(8.2.6) rj(x,z) = k \ -sin — dt, 

Jo r* 4r 

with k a certain constant. For points on the ship's course, where 
r = for some value t = t in the interval ^ t ^ T, this integral 
evidently does not exist. However, it has been shown in the preceding 
section that neighborhoods of such points can be ignored in calculating 
rj approximately provided that they are not points of stationary phase. 
This condition will be met in general, and hence we may imagine that 
a small interval about a point where r(t ) = has been excluded from 



WAVE PATTERN CREATED BY A MOVING SHIP 227 

the range of integration in case we wish the wave amplitude at a point 
on the ship's course. We write the integral in the form 

(8.2.7) r](x, z) = I y)(t)e i,p{t) dt, 

Jo 

and take the imaginary part. The function ip(t) and the phase cp(t) 
are given by 

(8.2.8) \p(t) = V 3 /r 4 

(8.2.9) <p(t) = gt 2 /4,r. 

We proceed to make the calculations called for in applying the 
stationary phase method. In the integral given by (8.2.7) no large 
parameter multiplying the phase is put explicitly in evidence; how- 
ever, from the discussion of the preceding section we know that the 
approximation will be good if the dimensionless quantity gR/4<c 2 , 
with R the distance from the ship, is large. It could also be verified 
that (8.2.6) would result if the integrations in (8.1.6) on fl and £ were 
first approximated by stationary phase followed by a re-introduction 
of the original variables. We therefore begin by calculating dcp/dt: 

dw s /2t t 2 dr\ 

(8.2.10 -f- = - — . 

dt 4\r r 2 dt] 

Hence the condition of stationary phase, dtpjdt = 0, leads to the im- 
portant relation 

(8.2.11) fU* 

dt t 

The quantity dr/dt is next calculated for the ship's course using 
(8.2.5); we find (cf. Figure 8.2.1): 



/It 

(8.2.12) r — 

V dt 






= — r • t = cr cos 6, 

in which c(t) is once more the speed of the ship. Thus 

dr 

(8.2.13) — = ccos0, 

dt 

which is a rather obvious result geometrically. Combining (8.2.11) 
and (8.2.13) yields the stationary phase condition in the form 

(8.2.14) r = id cos 6. 



228 



WATER WAVES 



We recall once more the significance of this relation: for a fixed point 
P(x, y) it yields those points Q { on C which are the sole points effective 
(within the order of the approximation considered) in creating the 
disturbance at P— the contributions from all other points being, in 
effect, cancelled out through mutual interference. It is helpful to intro- 
duce the term influence points for the points Q 4 determined in this 
way relative to a point P at which the surface elevation of the water 
is to be calculated. 

The last observation makes it possible to draw an interesting con- 
clusion at once from (8.2.14), which can be interpreted in the following 
way (cf. Figure 8.2.2): At point Q the speed c of the ship and t are 




Fig. 8.2.2. Points influenced by a given point Q 



known. The relation (8.2.14) then yields the polar coordinates (r, 6), 
with respect to Q, of all points P for which Q is the influence point in 
the sense of the stationary phase approximation. Such points P 
evidently lie on a circle with a diameter tangent to the course C of 
the ship at Q, and Q is at one end of the diameter. The center of the 
circle is located on the tangent line from Q in the direction toward 
which the ship moves (i.e. in the direction — t). We repeat that the 
points P on the circle just described are the only points for which Q 
is a point of stationary phase of the integral (8.2.7), and consequently 
the contribution of the impulse applied at Q vanishes (within the 
order considered by us) for all points except those on the circle. It 
now becomes obvious that the disturbance created by the ship does 
not affect the whole surface of the water, since only those points are 



WAVE PATTERN CREATED BY A MOVING SHIP 



229 



affected which lie on one or more of the circles of influence of all points 
Q on the ship's course. In other words, the surface waves created by 
the moving ship will be confined to the region covered by all the in- 
fluence circles, and thus to the region bounded by the envelope of this 
one-parameter family of curves. This makes it possible to construct 
graphically the outline of the disturbed region for any given course 
traversed at any given speed: one need only draw the circles in the 
manner indicated at a sufficient number of points Q and then sketch 
the envelope. Two such cases, one of them a straight course traversed 
at constant speed, the other a circular course, are shown in Figure 
8.2.3. In the case of the straight course it is clear that the envelope 





(a) (b) 

Fig. 8.2.3. Region of disturbance (a) Circular course (b) Straight course 

is a pair of straight lines; the disturbance is confined to a sector of 
semi-angle r given by r = arc sin 1/3 = 19°28', as one readily sees 
from Figure 8.2.3. This is already an interesting result: it says that 
the waves following the ship not only are confined to such a sector 
but that the angle of the sector is independent of the speed of the 
ship as long as the speed is constant. If the speed were not constant 
along a straight course, the region of disturbance would be bounded 
by curved lines, and its shape would also change with the time. It is, 
of course, not true that the disturbance is exactly zero outside the 
region of disturbance as we have defined it here; but rather it is 
small of a different order from the disturbance inside that region. 
The observations of actual moving ships bear out this conclusion in 
a quite startling way, as one sees from Figures 8.2.4 and 8.2.5. 
The discussion of the region of disturbance has furnished us with a 
certain amount of interesting information, but we wish to know a good 
deal more. In particular, we wish to determine the character of the 



230 



WATER WAVES 



wave pattern created by the ship and the amplitude of the waves. 
For these purposes a more thoroughgoing analysis is necessary, and it 
will be carried out later. 

In the special case of a straight course traversed at constant speed 
it is possible to draw quite a few additional conclusions through fur- 
ther discussion of the condition (8.2.14) of stationary phase. In the 
above discussion we asked for the points P influenced by a given 
point Q on the ship's course. We now reverse the question and ask for 




Fig. 8.2.4. Ships in a straight course 



WAVE PATTERN CREATED BY A MOVING SHIP 



231 



the location of all influence points Q i that correspond to a given point 
P. This question can be answered in our special case by making an- 
other simple geometrical construction (cf. Lamb [L.3], p. 435), as 
indicated in Figure 8.2.6. In this figure represents the location of 




Fig. 8.2.5a. A ship in a circular course 

the ship, P the point for which the influence points are to be deter- 
mined. The construction is made as follows: A circle through P with 
center on OP and diameter half the length of OP is constructed; its 
intersections with the ship's course are denoted by S 1 and S 2 . From 
the latter points lines are drawn to P and segments orthogonal to 
them at P are drawn to their intersections Q ± and Q 2 on the ship's 
course. The points Q x and Q 2 are the desired influence points. To prove 
that the construction yields the desired result requires only a verifica- 
tion that P does indeed lie on the influence circles determined by the 
points Q x and Q 2 in the manner explained above. Consider the point 



232 



WATER WAVES 



Q v for example. Since the angle S 1 PQ l = 90°, it follows that a 
circle with S ± Q ± as diameter contains the point P. The segments RS 1 
and PQ X are parallel since both are at right angles to S ± P; by con- 
sidering the triangle OPQ 1 one now sees that the segment OS ± is just 




Fig. 8.2.5b. Ships in curved courses 

half the length of OQ ± , and that is all that is necessary to show that the 
circle having S 1 Q 1 as diameter is the influence circle for Q v Thus there 
are in general two influence points or no influence points, the latter 
case corresponding to points P outside the influence region; the tran- 
sition occurs when P is on the boundary of the region of influence 
(i.e. when the circle of Figure 8.2.6 having PR as diameter is tangent 
to the course OQ 2 of the ship), and one sees that in this limit case the 
two influence points Q 1 and Q 2 coalesce. Consequently one might well 
expect that the amplitude of the waves at the boundary of the region 
of disturbance will be higher than at other places, and this phenome- 
non is indeed one of the prominent features always observed physical- 






WAVE PATTERN CREATED BY A MOVING SHIP 



233 



ly. In addition, the direction of the curves of constant phase — a wave 
crest, or trough, for example — can be determined graphically by the 
above construction: one expects these curves to be orthogonal to the 




Fig. 8.2.5c. Aircraft carriers maneuvering (from Life Magazine) 



lines PQ ± and PQ 2 drawn back from a point P to each of the points of 
influence corresponding to P. That this is indeed the case will be seen 
later, but it is evidently a consequence of the fact that the wave at P 
is the sum of two circular waves, one generated at Q 1 and the other at 



234 



WATER WAVES 



Q 2 . Thus we see that the wave pattern behind the ship is made up of 
two different trains of waves— another fact that is a matter of com- 
mon observation and which is well shown in Figures 8.2.4 and 8.2.5. 
We have been able to draw a considerable number of interesting and 




Fig. 8.2.6. Influence points corresponding to a given point 

basic conclusions of a qualitative character through use of the condi- 
tion of stationary phase (8.2.14). We proceed next to study analytic- 
ally the shape of the disturbed water surface by determining the 
curves of constant phase, and later on by determining the amplitude 
of the waves. To calculate the curves of constant phase it is convenient 
to express the basic condition (8.2.14) of stationary phase in other 
forms through introduction of the following quantity a, which has the 
dimension of length: 

2c 2 cH 2 

(8.2.15) a = — w = — . 

g 2r 

From (8.2.14) one then finds 

(8.2.16) ct = acosd, and 

(8.2.17) r = ±acos 2 6, 

as equivalent expressions of the stationary phase condition. 

It would be possible to calculate the curves of constant phase for 
any given course of the ship. We carry this out for the case of a cir- 
cular course (this case has been treated by L. N. Sretenski [S.15] ) and 
a straight course traversed at constant speed. The notation for the case 
of the circular course is indicated in Figure 8.2.7, which should be com- 
pared with Figure 8.2.1 . For the past position (x v %) of the ship we have 



WAVE PATTERN CREATED BY A MOVING SHIP 235 

2 




>X 



P(X,2) 

Fig. 8.2.7. Case of a circular course 

(8.2.18) J x x = R sin a 

j^ = .K(l — cos a) 
with 

(8.2.19) <x = ct/R. 

Here R is the turning radius of the ship, t the time required for it to 
travel from Q to 0, and c is the constant speed of the ship. The coor- 
dinates of the point P, where the disturbance created by the ship is 
to be found, are given by 

= x x — r cos (a + 0) 
r sin (a + d) 

in which r and 6 are the distance and angle noted on the figure. In 
these equations we replace x x and z 1 from (8.2.18) and make use of 
(8.2.17) to obtain 

a 



(8.2.20) 



(x = x x 
* = *! 



(8.2.21) 



R sin a — - cos 2 d cos (a -f 6) 



z = R(l — cos a) — - cos 2 Q sin (a + 6) . 



We wish to find the locus of points (x, z) such that the phase op remains 



236 



WATER WAVES 



fixed, i.e. such that the quantity a in (8.2.15) is constant (cf. the 
remarks following (8.2.9)). It is convenient to introduce the dimen- 
sionless parameter x through 

(8.2.22) x = a/R. 

One then finds that the angle a (cf. (8.2.19)) is given by 

(8.2.23) vl = xcos6, 

through use of (8.2.16). In terms of these quantities the relations 
(8.2.21) can be put in the following dimensionless form: 



(8.2.24; 



xjR = sin (x cos Q) 



- cos 2 6 cos (6 + x cos 6) 



z/R = 1 — cos (x cos 6) — - cos 2 6 sin (6 + x cos 6). 



These equations furnish the curves of stationary phase in terms of 6 
as parameter. Each fixed value of x furnishes one such curve, since 
fixing x (for a fixed turning radius R) is equivalent to fixing the phase 
<p. In Figure 8.2.8 a few curves of constant phase, as well as the 




Fig. 8.2.8. Wave crests for a circular course 

outline of the region of disturbance, as calculated from (8.2.24), are 
shown; the successive curves differ by 2n in phase. These curves should 
be compared with the photographs of actual cases given in Figures 
8.2.4 and 8.2.5. One sees that the wave pattern is given correctly by 
the theory, at least qualitatively. The agreement between theory and 
observation is particularly striking in view of the manner in which 
the action of a ship has been idealized as a moving pressure point. In 
particular there are two distinct sets of waves apparent, in conformity 
with the fact that we expect each point in the disturbed region to 



WAVE PATTERN CREATED BY A MOVING SHIP 



237 



correspond to two influence points: one set which seems to emanate 
from the ship's bow, and another set which is arranged roughly at 
right angles to the ship's course. These two systems of waves are called 
the diverging and the transverse systems, respectively. 

From (8.2.24) we can obtain the more important case of the ship 
waves for a straight course by letting R -> go while « -> in such a 
way that Rx -+ a (cf. (8.2.22)). The result is 



(8.2.25) 



(2cos0 — cos 3 0) 



z = — - cos 2 6 sin 
2 



for the curves of constant phase. In Figure 8.2.9 the results of cal- 
culations from these equations are shown. These should once more be 
compared with Figure 8.2.4, which shows an actual case. Again the 
agreement is striking in a qualitative way. Actually, the agreement 




Fig. 8.2.9. Wave crests for a straight course 



would be still better if the two systems of waves — the diverging and 
transverse systems — had been drawn in Figure 8.2.9 with a relative 
phase difference: the photograph indicates that the crests of the two 
systems do not join with a common tangent at the boundary of the 
region of disturbance. We shall see shortly that a closer examination 
of our approximate solution shows the two systems of waves to have a 
phase difference there. It is worth while to verify in the present case a 
general observation made earlier, i.e. that the curves of constant phase 



238 



WATER WAVES 



are orthogonal to the lines drawn back to the corresponding influence 
points. One finds from (8.2.25): 



(8.2.26) 



dx 
dd 
dz 
dd 



(3 sin 2 6 - 1 ) sin 6 



(3 sin 2 — 1) cos 6. 



Hence dz/dx = — 1/tan 6, which (cf. Figure 8.2.10) means that the 
curves of constant phase are indeed orthogonal to the lines drawn to 




Fig. 8.2.10. Construction of curves of constant phase 

the influence points. The values 6 = 6* at which 3 sin 2 — 1=0 
are singular points of the curves; they correspond to points P at the 
boundary of the influence region where the influence points Q x and 
Q 2 coincide. Evidently there are cusps at these points. One sees also 
that the diverging set of waves (for z > 0, say) is obtained when 6 
varies in the range 6* ^ 6 ^ Tt/2, while the transverse waves corres- 
pond to values of 6 in the range 0^6^ 6*. In addition, we observe 
that to any point on the ship's course there corresponds (for = 0°) 
only one influence point (of type Q 2 ) and it does not coincide with the 
point P. (One sees, in fact, that the diverging wave does not occur on 
the ship's course.) This is a fact that is needed to justify the applica- 
tion of the method of stationary phase to points on the ship's course, 
as we have remarked earlier in this section (cf. also the preceding 
section). 

In order to complete our discussion we must consider the amplitude 
of the surface waves, as given by our approximation, as well as the 
shape of the curves of constant phase. To this end we must calculate 
99 and d 2 (p/dt 2 (and even d 3 (p/dt 3 ) for such values of t as satisfy the 



WAVE PATTERN CREATED BY A MOVING SHIP 



239 



stationary phase condition dcp/dt = 0, as we know from the discussion 
of section 6.5 and section 6.8. From (8.2.10) we find easily 



(8.2.27 



dt 2 



g_ L _ *_ &r\ 

2r \ 2r dt 2 ) 



in view of (8.2.11). We shall also need the value of d 3 (p/dt* at points 
such that dcpjdt = d 2 (p/dt 2 = 0; it is readily found to be given by 

d*<p _ gt 2 d*r 



(8.2.28) 



We wish to express our results in terms of the parameter instead of 
t. Since drjdt = c cos 6 from (8.2.13) we have 



(8.2.29) 



d*r 
df 2 



c sin0 



dd 

~dt 



with c, the speed of the ship, now assumed to be constant. In order 
to calculate ddjdt we introduce the angles ft and r indicated in Figure 
8.2.11. W T e have d = n — (/? + t), and hence 



Z'i 




P(x,z) 





Fig. 8.2.11. The angles p and r 


dd 

Hi 


/ dp dr\ds /dp dx 

\ ds ds J dt \ds ds 



(8.2.30) 



in which s refers to the arc length of C. But dr/ds = 1/R, with R the 
radius of curvature of C; and since p = arc tan (z — %\)l(x - ^i) we 
find 



(8.2.31 



ds 



(X 



dz 1 
ds 



(z 



dx x 
~ds~ 



sin 6 
r 



240 WATER WAVES 

since the quantity in the square brackets is the vector product of r 
and t/|t|. The expression for d 2 (p/dt 2 given by (8.2.27) can now be 
expressed in terms of and r as follows: 



d*w g 

(8.2.32 —L = — 



2r 



1-2 tan 2 6 — - sin 
R 



1-3 sin 2 — — sin (1 — sin 2 0) 
R 



cos 2 



as one can easily verify. The quantity a is defined by (8.2.15), and 
the relation (8.2.17), in addition to those immediately above, has 
been used. The points on the boundary of the region of disturbance 
could be determined analytically, as follows: the set of all influence 
points is the one-parameter family of circles given by dcp/dt = %(%,z,t) 
= 0, and the region of disturbance is bounded by the envelope of 
these circles, i.e. by the points at which d 2 cpjdt 2 = d%\dt = in 
addition to % = 0. In the case of a straight course traversed at con- 
stant speed, for example, we see from (8.2.32) for R = oo that 
then has the value 0* given by 1 — 3 sin 2 = — a result found above, 
where the value = 0* also was seen to characterize cusps on the loci 
of constant phase. From the form of the relation (8.2.32) one can con- 
clude that the only courses for which the pattern of waves behind the 
ship follows it without change (i.e. follows it like a rigid body) are 
those for which R = const.; and thus only the straight and the cir- 
cular courses have this property. 

Finally, we have to consider the amplitude r)(x, z) of the waves given 
by our approximate solution. The contribution of a point t of sta- 
tionary phase to (8.2.7) is given by (cf. (6.5.2)): 



(8.2.33 



rj(x,z) =y(r, 0) — eV O 

\\(p"(r, 0)|/ 



in which (r, 0) are polar coordinates which locate the point of sta- 
tionary phase on the course C relative to the point (tc, z) (cf. Figure 
8.2.1). The sign of the term i jt/4 is to be taken the same as that of 
cp" = d 2 cp/dt 2 . In principle, the surface elevation can be calculated for 
any course, but the results are not very tractable except for the 
simplest case; i.e. a straight course. We confine our discussion of 
amplitudes, therefore, to this case in what follows. From (8.2.32) we 
have 






WAVE PATTERN CREATED BY A MOVING SHIP 241 

d 2 w g /l - 3 sin 2 0\ 

8.2.34 —*- = — |. 

dt 2 2r \ cos 2 J 

We know that there are two values of — call them d x and 2 — at each 
point in the disturbed region for which d(p/dt = 0: one belonging for 
^ X ^ 0* = arc sin l/\/3 to the transverse system, the other for 
0* ^ 6 2 < ti/2 to the diverging system of waves. In the former case 
d 2 <p/dt 2 is positive; in the latter case negative. (At the boundary of 
the region of disturbance, where op" = 0, the formula (8.2.33) is not 
valid, as we know. This case will be dealt with later.) For points in 
the interior of the region of disturbance we have, therefore, 

V( r i» °i) 



(8.2.35) rj(se, z) C^. V2ti 



+ W r 2> 2 ) 



1 
VV'(r 2 ,0 a )| 



i(*P(r 2 , 2 )-?)' 



Since r i = \ ct t cos 6 { , r i = \a t cos 2 Z -, a t = 2c 2 (p i /g = cH\j2r i9 and 
ip = k tyr% (cf. (8.2.15), (8.2.8)) at the points of stationary phase, 
we may write (8.2.35) in the form 



(8.2.36) r){x,z)~2y%7i ll2 /c*g 112 — e l \ 

I V|l - 3 sin 2 X 



sec 3 6 1 



. {sa x 



sec 3 f 



a\ V 1 1 - 3 sin 2 2 



./goa ^\ 



The two systems of waves are thus seen, as was stated above, to have 
a relative phase difference of 7i/2 at any point where a x = a 2 . Their 
amplitudes die out like 1/V a i on g om g away from the ship, and that 
means that they die out like the inverse square root of the distance 
from the ship. The wave amplitudes of both systems of waves become 
infinite according to these formulas for = 0*, i.e. for points at the 
boundary of the disturbed region, but the asymptotic formula (8.2.33) 
is not valid at such points since 99" = there. We shall consider these 
points in a moment. The diverging system also has infinite amplitude 
for 2 = 7t/2, but this corresponds to the origin, and the infinite am- 
plitude there results from our assumption of a moving point impulse 
as a model for our ship. 

To determine the amplitude of the waves along the boundary of the 
disturbed region, we must calculate the value of d z (p/dt 3 at such points 
in order to evaluate the appropriate term in (6.5.2). (The problem of 



242 WATER WAVES 

the character of the waves in this region has been treated by Hogner 
[H.13].) By differentiating (8.2.29) after replacing dd/dt by c sin 0/r 
(cf. (8.2.30) and (8.2.31) for R = oo), one finds readily 

d 3 r c 3 cos 6 sin 2 6 

(8.2.37 — = , 

dt* r 2 

and from (8.2.28) in combination with r = \ct cos 0,r = \a cos 2 6: 

d 3 (p 4<gc sin 2 6 



(8.2.38) 



dt 3 a 2 cos 5 d 



The amplitude of the waves along the boundary of the disturbed 
region is given by (cf. (6.5.2)): 

(8.2.39) n(x,z)~ ^y>(6*) (—J—-\ i e *<•*>, 



■n(x, z) ~ -SMw(6*) ( V 

A V3 * \\<P'"(6*)\) 



with all functions evaluated for = 0* = arc sin l/\/3. The final 
result is 

(8.2.40) r] ^ -^-exp {iga/2c 2 }. 



a 



1/3 



The quantity k ± is a certain constant. We observe that the wave am- 
plitudes now die out like 1/a 113 instead of like l/a 1/2 , as they do in the 
interior of the disturbed region; i.e. the wave amplitudes are now of a 
different, and higher, order of magnitude. As we have seen in all of 
our illustrations of ship waves, the wave amplitudes are quite notice- 
ably higher along the boundary of the disturbed region. The phase 
also differs now by tt/4 from the former values. On some of the photo- 
graphs (cf. especially Fig. 8.2.4), there is some evidence of a rather 
abrupt change of phase in the region of the boundary, though it may 
be that one should interpret this effect as due rather to the finite 
dimensions of the actual ship, which then acts as though several 
moving point sources were acting simultaneously. 

In the treatment of the present problem by A. S. Peters [P.4] 
mentioned in the preceding section, the complete asymptotic develop- 
ment of the solution was obtained. 

The above developments hold only for the case of a point impulse 
moving on the surface of water of infinite depth. It has some interest 
to point out that there are considerable differences in the results if 
the depth of the water is finite. Havelock [H.8] has carried out the 
approximation to the solution by the method of stationary phase for 



WAVE PATTERN CREATED BY A MOVING SHIP 



243 



the case of constant finite depth, with the following general results: 
1) If the speed c of the ship and the depth h satisfy the inequality 
c 2 lgh < 1, the general pattern of the waves is much the same as for 
water of infinite depth except that the angle of the sector within 
which the main part of the disturbance is found is now larger than 
for water of infinite depth. 2) If c 2 /gh > 1 holds, the system of trans- 
verse waves no longer occurs, but the diverging system is found. 



1 ^ 


. -^ -.. ■ - ./ r 




r^^?v^; 




*>& ^^^^■^tM 


vqV.'V 1 -* /"'*> 


■ ■ T ifli ■17 




H0m8DwS^*w v- '-;■-*?- 


■LV' " "" ..< ■ » ■ 


■EiGfS^VJp -A:-'. 




■EMS' * K-%_ . 




K ' ; i| 


fe^/fc' . . • ■ ■ ", •■ ,' 




vtEr" 


- ■->* "•■- !■ ',. si 




-^;- -« ■-' _ J. 1 -fc 


-j, ■ T»sir ., ,. t -.'; 


'>tV^)^J^H^» *^ 


^S^Si^* ''"■'■' 


■ -/^^Hy 




■ * ,,, *^G»S3"2SE5**'' 


t tvlwv 


^•.' " : "i^sSSJSSji 


* V\ i^^N^V 


.£*■'** ' •^'^'^ss^is 


I ( V " ■>>>%<>; 


- ''^^SEBlHN 


1*1% "V* V * ■• 


• "^f^^i™ 


1 A k ** N \ 




■ S "■ 1 % ^k 


^'SOH 


1 U W % -% ; ■ ^V 


.^^asKM 


••ft I) M v,* *> \ 


"- '!** : "' ' JH jj 


idr > F iJ *• ; '«' *J " 


' "'""" ~" jBI 


/ ////* '";.'•'" 




*"j£"' J ' ./' ) 


HB* "*'^!niEL-' ■ Wm^^mEmM 


&PP& 5 •"■•" r W J 


?|W^»»Z^ ^w^rw^'*ie»*S to >^ 


-~ hS / 


^«y!!^PJ>Mfci|i^i wW^BBB^iiifcii**^ , , 


" »»*• ■* .'^ 


^^^i*T**T^5JfS 


J ^^^afc^l 




SSfeSStS 





Fig. 8.2.12. Speed boat in shallow water 



Figure 8.2.12 is a photograph of a speed boat creating waves, presum- 
ably in shallow water, in view of the difference in the wave pattern 
when compared with Fig. 8.2.4. Finally, if c 2 /gh = 1 (i.e. for the case 
of the critical speed), the method of stationary phase yields no rea- 
sonable results; that this should be so is perhaps to be understood in 
the light of the discussion of the corresponding two-dimensional 
problem in Chapter 7.4. 



CHAPTER 9 

The Motion of a Ship, as a Floating Rigid Body, in a 

Seaway 

9.1. Introduction and summary 

The purpose of this chapter is to develop a mathematical theory for 
the motion of a ship, to be treated as a freely floating rigid body under 
the action of given external forces (a propeller thrust, for example), 
under the most general conditions compatible with a linear theory and 
the assumption of an infinite ocean.* This of course requires the 
amplitude of the surface waves to be small and, in general, that the 
motion of the water should be a small oscillation near its rest position 
of equilibrium; it also requires the ship to have the shape of a thin 
disk so that it can have a translatory motion with finite velocity and 
still create only small disturbances in the water. In addition, the mo- 
tion of the ship itself must be assumed to consist of small oscillations 
relative to a motion of translation with constant velocity. Within 
these limitations, however, the theory presented is quite general in 
the sense that no arbitrary assumptions about the interaction of the 
ship with the water are made, nor about the character of the coupling 
between the different degrees of freedom of the ship, nor about the 
waves present on the surface of the sea: the combined system of ship 
and sea is treated by using the basic mathematical theory of the 
hydrodynamics of a non-turbulent perfect fluid. For example, the 
theory presented here would make it possible in principle to deter- 
mine the motion of a ship under given forces which is started with 
arbitrary initial conditions on a sea subjected to given surface pres- 
sures and initial conditions, or on a sea covered with waves of pre- 
scribed character coming from infinity. 

It is of course well known that such a linear theory for the non- 
turbulent motion of a perfect fluid, complicated though it is, still does 
not contain all of the important elements needed for a thoroughgoing 
discussion of the practical problems involved. For example, it ignores 

* The presentation of the theory given here is essentially the same as that 
given in a report of Peters and Stoker [P.7]. 

245 



246 



WATER WAVES 



the boundary-layer effects, turbulence effects, the existence in general 
of a wake, and other important effects of a non-linear character. Good 
discussions of these matters can be found in papers of Lunde and Wig- 
ley [L.18], and Havelock [H.7]. Nevertheless, it seems clear that an 
approach to the problem of predicting mathematically the motion of 
ships in a seaway under quite general conditions is a worthwhile enter- 
prise, and that the problem should be attacked even though it is 
recognized at the outset that all of the important physical factors can 
not be taken into account. In fact, the theory presented here leads at 
once to a number of important qualitative statements without the 
necessity of producing actual solutions— for example, we shall see 
that certain resonant frequencies appear quite naturally, and in 
addition that they can be calculated solely with reference to the mass 
distribution and the given shape of the hull of the ship. Interesting 
observations about the character of the coupling between the various 
degrees of freedom, and about the nature of the interaction between 
the ship and the water, are also obtained simply by examining the 
equations which the theory yields. 

In order to describe the theory and results to be worked out in 
later sections of this chapter, it is necessary to introduce our notation 
and to go somewhat into details. In Fig. 9.1.1 the disposition of two of 
the coordinate systems used is indicated. The system (X, Y, Z) is a 





Fig. 9.1.1. Fixed and moving coordinate systems 

system fixed in space with the X, Z-plane in the undisturbed free 
surface of the water and the F-axis vertically upward. A moving 
system of coordinates (cc, y, z) is introduced; in this system the x, z- 
plane is assumed to coincide always with the X, Z-plane, and the 
?/-axis is assumed to contain the center of gravity (abbreviated to e.g. 
in the following) of the ship. The course of the ship is fixed by the 
motion of the origin of the moving system, and the #-axis is taken along 



THE MOTION OF A SHIP IN A SEAWAY 



247 



the tangent to the course. It is then convenient to introduce the 
speed s(t) of the ship in its course: the speed s(t) is simply the magni- 
tude of the vector representing the instantaneous velocity of this point. 
At the same time we introduce the angular speed co(t) of the moving 
system relative to the fixed system: one quantity fixes this rotation 
because the vertical axes remain always parallel. The angle cc(t) 
indicated in Fig. 9.1.1 is defined by 



(9.1.1 



K(f) 



J o 



co(t) dt, 



implying that t = corresponds to an instant when the <z-axis and 
X-axis are parallel. In order to deal with the motion of the ship as a 
rigid body it is convenient, as always, to introduce a system of coor- 
dinates fixed in the body. Such a system (x\ y', z') is indicated in 
Fig. 9.1.2. The x\ i/'-plane is assumed to be in the fore-and-aft plane 




^rnvj^zm? 



(b) 



Fig. 9.1.2a, b. Another moving coordinate system 



of symmetry of the ship's hull, and the ?/'-axis is assumed to contain 
the e.g. of the ship. The moving system^', y', z') is assumed to coin- 
cide with the (x, y, z) system when the ship and the water are at rest 
in their equilibrium positions. The e.g. of the ship will thus coincide 
with the origin of the (x', y', z') system only in case it is at the level 
of the equilibrium water line on the ship; we therefore introduce the 
constant y' c as the vertical coordinate of the e.g. in the primed coor- 
dinate system. 

The motion of the water is assumed to be given by a velocity poten- 
tial 0(X, Y, Z; t) which is therefore to be determined as a solution 
of Laplace's equation satisfying appropriate boundary conditions at 
the free surface of the water, on the hull of the ship, at infinity, and 
also initial conditions at the time t = 0. The boundary conditions on 
the hull of the ship clearly will depend on the motion of the ship, 



248 WATER WAVES 

which in its turn is fixed, through the differential equations for the 
motion of a rigid body with six degrees of freedom, by the forces acting 
on it —including the pressure of the water — and its position and veloc- 
ity at the time t = 0. As was already stated, no further restrictive 
assumptions except those needed to linearize the problem are made. 
Before discussing methods of linearization we interpolate a brief 
discussion of the relation of the theory presented here to that of other 
writers who have discussed the problem of ship motions by means of 
the linear theory of irrotational waves. The subject has a lengthy 
history, beginning with Michell in 1898, and continuing over a long 
period of years in a sequence of notable papers by Havelock, starting 
m 1909. This work is, of course, included as a special case in what is 
presented here. Extensive and up-to-date bibliographies can be found 
in the papers by Weinblum [W.3] and Lunde [L.19]. Most of this work 
considers the ship to be held fixed in space while the water streams 
past; the question of interest is then the calculation of the wave 
resistance in its dependence on the form of the ship. Of particular 
interest to us here are papers of Krylov [K.20], St. Denis and Wein- 
blum [S.l], Pierson and St. Denis [P. 9] and Haskind [H.4], all of 
whom deal with less restricted types of motion. Krylov seeks the 
motion of the ship on the assumption that the pressure on its hull 
is fixed by the prescribed motion of the water without reference to 
the back effect on the motion of the water induced by the motion 
of the ship. St. Denis and Weinblum, and Pierson and St. Denis, 
employ a combined theoretical and empirical approach to the prob- 
lem which involves writing down equations of motion of the ship 
with coefficients which should be in part determined by model ex- 
periments; it is assumed in addition that there is no coupling be- 
tween the different degrees of freedom involved in the general mo- 
tion of the ship. Haskind attacks the problem in the same degree 
of generality, and under the same general assumptions, as are made 
here; in the end, however, Haskind derives his theory completely only 
in a certain special case. Haskind's theory is also not the same as the 
theory presented here, and this is caused by a fundamental difference 
in the procedure used to derive the linear theory from the underlying, 
basically nonlinear, theory. Haskind develops his theory by assuming 
that he knows a priori the relative orders of magnitude of the various 
quantities involved. The problem is attacked in this chapter by a 
formal development with respect to a small parameter (essentially a 
thickness-length ratio of the ship); in doing so every quantity is 



THE MOTION OF A SHIP IN A SEAWAY 249 

developed systematically in a formal series (for a similar type of 
discussion see F. John [J. 5] ). In this way a correct theory should be 
obtained, assuming the convergence of the series — and there would 
seem to be no reason to doubt that the series would converge for 
sufficiently small values of the parameter. Aside from the relative 
safety of such a method— purchased, it is true, at the price of making 
rather bulky calculations — it has an additional advantage, i.e., it 
makes possible a consistent procedure for determining any desired 
higher order corrections. It is not easy to compare Haskind's theory 
in detail with the theory presented here. However, it can be stated 
that certain terms, called damping terms by Haskind, are terms that 
would be of higher order than any of those retained here. A more 
precise statement on this point will be made later. 

One of the possible procedures for linearizing the problem begins 
by writing the equation of the hull of the ship relative to the coordinate 
system fixed in the ship in the form 

(9.1.2) z' = ±fih(ai',y'), z'>0, 

with f$ a small dimensionless parameter.* This is the parameter with 
respect to which all quantities will be developed. In particular, the 
velocity potential @(X, Y, Z; t; /?) = (p(x, y, z; t; @) is assumed to 
possess the development 

(9.1.3) cp(x, y, z; t; ft) = /%(#, y, z; t) + fi 2 (p 2 (x, y, z; t) + . . . . 

The free surface elevation rj(%, z; t; (3) and the speed s(t; @) and angu- 
lar velocity co(t; ft) (cf. (9.1.1)) are assumed to have the developments 

(9.1.4) rj(x, z; t; 0) = fa(m, z; t) + ^{x, z; t) + . . . , 

(9.1.5) s(t; /?) = s (t) + fait) +..-, 

(9.1.6) <o(t; 0) = co (t) + pco^t) +... . 

Finally, the vertical displacement y c (t) of the center of gravity and 
the angular displacements** V 6 2 , 3 of the ship with respect to the 
x, y, and z axes respectively are assumed given by 

* It is important to consider other means of linearization, and we shall discuss 
some of them later. However, it should be said here that the essential point is 
that a linearization can be made for any body having the form of a thin disk: 
it is not at all essential that the plane of the disk should be assumed to be vertical, 
as we have done in writing equation (9.1.2). 

** Since we consider only small displacements of the ship relative to a uniform 
translation, it is convenient to assume at the outset that the angular displacement 
can be given without ambiguity as a vector with the components ls 2 , 6 3 relative 
to the x, y, z-coordinate system. 



250 WATER WAVES 

(9.1.7) e t (t ; 0) = pe a (t) + pOaM + ■ ■ • . » = i. 2, 3, 

(9.1.8) y,(«; fi)- V ', = fa(t) + /»V,W + • • • • 

These relations imply that the velocity of the water and the 
elevation of its free surface are small of the same order as the "slender- 
ness parameter" f$ of the ship. On the other hand, the speed s(t) of the 
ship is assumed to be of zero order. The other quantities fixing the 
motion of the ship are assumed to be of first order, except for a>(t), 
but it turns out in the end that co (t) vanishes so that co is also of first 
order. The quantity y' c in (9.1.8) was defined in connection with the 
description of Fig. 9.1.2; it is to be noted that we have chosen to 
express all quantities with respect to the moving coordinate system 
(x, y, z) indicated in that figure. The formulas for changes of coordi- 
nates must be used, and they also are to be developed in powers of 
{$; for example, the equation of the hull relative to the (x, y, z) co- 
ordinate system is found to be 

z + pe n x - pe n (y - y' c ) - Ph(x, y) + . . . = 

after developing and rejecting second and higher order terms in {$. 
In marine engineering there is an accepted terminology for describ- 
ing the motion of a ship; we wish to put it into relation with the no- 
tation just introduced. In doing so, the case of small deviations from 
a straight course is the only one in question. The angular displace- 
ments are named as follows: d 1 is the rolling, 6 2 + a is the yawing, and 
d 3 is the pitching oscillation. The quantity fis^t) in (9.1.5) is called 
the surge (i.e., it is the small fore-and-aft motion relative to the finite 
speed s (t) of the ship, which turns out to be necessarily a constant), 
while y c — y' c fixes the heave. In addition there is the sidewise dis- 
placement dz referred to as the sway; this quantity, in lowest order, 
can be calculated in terms of s (t) and the angle oc defined by (9.1.1) 
in terms of oj(t) as follows: 



(9.1.9) dz = s ol = fis Q (o x (t)dl 



o w i< 

J o 



since oj (t) turns out to vanish. 

In one of the problems of most practical interest, i.e. the problem 
of a ship that has been moving for a long time (so that all transients 
have disappeared) under a constant propeller thrust (considered to be 
simply a force of constant magnitude parallel to the keel of the ship) 



THE MOTION OF A SHIP IN A SEAWAY 251 

into a seaway consisting of a given system of simple harmonic progres- 
sing waves of given frequency, one expects that the displacement com- 
ponents would in general be the sum of two terms, one independent of 
the time and representing the displacements that would arise from 
motion with uniform velocity through a calm sea, the other a term 
simple harmonic in the time that has its origin in the forces arising 
from the waves coming from infinity. On account of the symmetry of 
the hull only two displacements of the first category would differ 
from zero: one the vertical displacement, i.e. the heave, the other the 
pitching angle, i.e. the angle 6 3 . The latter two displacements apparent- 
ly are referred to as the trim of the ship. In all, then, there would be 
in this case nine quantities to be fixed as far as the motion of the ship 
is concerned: the amplitudes of the oscillations in each of the six 
degrees of freedom, the speed s , and the two quantities determining 
the trim. A procedure to determine all of them will next be outlined. 
We proceed to give a summary of the theory obtained when the 
series (9.1.2) to (9.1.8) are inserted in all of the equations fixing the 
motion of the system, which includes both the differential equations 
and the boundary conditions, and any functions involving fi are in 
turn developed in powers of /?. For example, one needs to evaluate <p x 
on the free surface y = r\ in order to express the boundary conditions 
there; one calculates it as follows (using (9.1.3) and (9.1.4)): 

(9.1.10) <p x (x, r), z; t; /?) = fl[(p lx (x, 0, z; t) + T]<p lxy (x, 0, z; t) + ...] 

= (tyuifa °> *J *) + P 2 [yi<Plxy(v> °> Z '> t) + <P2x(®> o, z; t)] +... . 

We observe the important fact — to which reference will be made 
later — that the coefficients of the powers of ft are evaluated at y = 0, 
i.e. at the undisturbed equilibrium position of the free surface of the 
water. In the same way, it turns out that the boundary conditions 
for the hull of the ship are automatically to be satisfied on the vertical 
longitudinal mid-section of the hull. The end result of such calcula- 
tions, carried out in such a way as to include all terms of first order in 
f} is as follows: The differential equation for (p ± is, of course, the La- 
place equation: 

(9-1-H) <Pixx + (Piy y + (p lzz = 

in the domain y < 0, i.e. the lower half-space, excluding the plane 
area A of the x, ?/-plane which is the orthogonal projection of the 



252 WATER WAVES 

hull (cf. Fig. 9.1.2b), in its equilibrium position, on the x, ?/-plane. 
The boundary conditions on cp ± are 

<Pu = - s o(K - 21 ) — (o) x + Q 21 )x + dn(y - y'\ on A + 



(9.1.12; 

<Pu = s Q (h x + d 21 ) - {co x + Q 21 )x + $ n (y — y e ), on A. 

in which A + and ^4_ refer to the two sides z = + and 2 = 0_ of the 
plane disk A. The boundary conditions on the free surface are 

(9.1.13) (-*h+*rfV--*" = Bty = 0. 
I — Piv — Wi« + *7i* = ° 

The first of these results from the condition that the pressure vanishes 
on the free surface, the second arises from the kinematic free surface 
condition. If s , co lf 6 21 , and # n were known functions of t, these boun- 
dary conditions in conjunction with (9.1.11) and appropriate initial 
conditions would serve to determine the functions cp x and rj 1 uniquely; 
i.e. the velocity potential and the free surface elevation would be 
known. Of course, the really interesting problems for us here are those 
in which the quantities ,s , co 1? 21 , and 1V referring to the motion of 
the ship, are not given in advance but are rather unknown functions 
of the time to be determined as part of the solution of the boundary 
problem. In principle, one method of approach would be to apply the 
Laplace transform with respect to the time t to (9.1.11), (9.1.12), and 
(9.1.13)— of course taking account of initial conditions at the time 
t = — and then to solve the resulting boundary value problem 
for the transform (p^x, y, z; a) regarding s and the transforms co 1 (cr), 
21 (cr), and § 11 ((t) as parameters. However, for the purposes of this 
introduction it is better to concentrate on the most important special 
case (already mentioned above) in which the ship has a motion of 
translation with uniform speed combined with small simple harmonic 
oscillations of the ship and the sea having the same frequency.* In 
this case we write the velocity potential y^x, y, z; t), the surface 
elevation rj v and the other dependent quantities in the form 

( <p ± (x, y, z; t) = yj (x, y, z) + y ± (x, y, z)e iat 

(9.1.14) rj^x, z; t) = H (x, z) + H ± (x, z)e iat 

{ co, = Q x e ia \ 6 n = V"°' 02i = @2ie iat - 
The functions yj and yj ± are of course both harmonic functions. We 
expect the functions cp x and r] x to have time-independent components 

* It can be seen, however, that the discussion which follows would take much 
the same course if more general motions were to be assumed. 



THE MOTION OF A SHIP IN A SEAWAY 253 

due to the forward motion of the ship; certainly they would appear 
in the absence of any oscillatory components due, say, to a wave train 
in the sea. Upon insertion of these expressions in equations (9.1.12). 
and (9.1.13) we find for ip the conditions: 

( ip 0z = T s h x , on A ± , 
(9.1.14) gH - s Q y) 0a 

and for \p x the conditions 

( — Viz = — s o 21 ^ r {Q 1 -\-iG0 21 )x — ioG^y—y'c) on A ± 
(9.1.14)! — gH t + s y lx - ia\p 1 = 



:) 



at y = 0, 



at y = 0. 
Wiy — s Q H lx + ioH 1 = 

We observe, in passing, that ip satisfies the same boundary conditions 
as in the classical Michell-Havelock theory. A little later we shall see, 
in fact, that the wave resistance is indeed independent of all compo- 
nents of the motion of the ship (to lowest order in f$, that is) except its 
uniform forward motion with speed s , and that the wave resistance 
is determined in exactly the same way as in the Michell-Havelock 
theory. We continue the description of the equations which determine 
the motion of the ship, and which arise from developing the equations 
of motion with respect to f} and retaining only the terms of order /? 
and /? 2 . (We observe that it is necessary to consider terms of both 
orders.) In doing so the mass M of the ship is given by M = Mjfi, 
with M 1 a constant, since we assume the average density of the ship 
to be finite and its volume is of course of order 0. The moments of 
inertia are then also of order /?. The propeller thrust is assumed to be 
a force of magnitude T acting in the ^'-direction and in the x\ y'- 
plane at a point whose vertical distance from the e.g. is — /; the thrust 
T is assumed to be of order /? 2 , since the mass is of order f$ and accelera- 
tions are also of order p.* The propeller thrust could also, of course, 
be called the wave resistance. 

The terms of order f} yield the following conditions: 

* We have in mind problems in which the motion of the ship is a small deviation 
from a translatory motion with uniform finite speed. If it were desired to study 
motions with finite accelerations — as would be necessary, for example, if the 
ship were to be considered as starting from rest — it would clearly be necessary 
to suppose the development of the propeller thrust T to begin with a term of first 
order in /?, since the mass of the ship is of this order. In that case, the motion 
of the ship at finite speed and acceleration would be determined independently 
of the motion of the water: in other words, it would be conditioned solely by the 
inert mass of the ship and the thrust of order /?. 



254 


WATER WAVES 


(9.1.15) 


s = 0, 


(9.1.16) 


2 Q g f phdA = MJg, 


(9.1.17) 


j xfihdA = 0, 


(9.1.18) 


[(<Plt - Wla)]* dA = °> 


(9.1.19) 


[x((p lt - s cp lx )] + _ dA = 0, 


(9.1.20) 


M<Pit - Sq<Pix)T_ dA = 0. 



The symbol [ ] + _ occurring here means that the jump in the quan- 
tity in brackets on going from the positive to the negative side of the 
projected area A of the ship's hull is to be taken. The variables of in- 
tegration are x and y. The equation (9.1.15) states that the term of 
order zero in the speed is a constant, and hence the motion in the 
^-direction is a small oscillation relative to a motion with uniform 
velocity. (This really comes about because we assume the propeller 
thrust T to be of order /? 2 . ) Equation (9.1.16) is an expression of the 
law of Archimedes: the rest position of equilibrium must be such that 
the weight of the ship just equals the weight of the water it displaces. 
Equation (9.1.17) expresses another law of equilibrium of a floating 
body, i.e. that the center of buoyancy should be on the same vertical 
line as the center of gravity of the ship. The remaining three equations 
(9.1.18), (9.1.19), and (9.1.20) in the group serve to determine the dis- 
placements 1V 6 21 , and co v which occur in the boundary condition 
(9.1.12) for the velocity potential <p v In the special case we consider 
(cf. (9.1.14)) we observe that these three equations would determine 
the values of the constants Q v (9 n , and 21 (the complex amplitudes 
of certain displacements of the ship) which occur as parameters in the 
boundary conditions for the harmonic function ip^x, y, z) given in 
(9.1.14),. 

We are now able to draw some interesting conclusions. Once the 
speed s is fixed, it follows that the problem of determining the har- 
monic function cp 1 is completely formulated through the equations 
(9.1.14), (9.1.14) , (9.1.14)!, and (9.1.18) to (9.1.20) inclusive (to- 
gether with appropriate conditions at oo). In other words, the motion 
of the water, which is fixed solely by 9^, is entirely independent of the 



THE MOTION OF A SHIP IN A SEAWAY 255 

pitching displacement 31 (t), the heave y ± (t), and the surge s^t), i.e. 
of all displacements in the vertical plane except the constant forward 
speed s . A little reflection, however, makes this result quite plausible: 
Our theory is based on the assumption that the ship is a thin disk 
disposed vertically in the water, whose thickness is a quantity of 
first order. Hence only finite displacements of the disk parallel to 
this vertical plane could create oscillations in the water that are of 
first order. On the other hand, displacements of first order of the disk 
at right angles to itself will create motions in the water that are also 
of first order. One might seek to describe the situation crudely in the 
following fashion. Imagine a knife blade held vertically in the water. 
Up-and-down motions of the knife evidently produce motions of the 
water which are of a quite different order of magnitude from motions 
produced by displacements of the knife perpendicular to the plane of 
its blade. Stress is laid on this phenomenon here because it helps to 
promote understanding of other occurrences to be described later. 
The terms of second order in ($ yield, finally, the following conditions : 

(9.1.21) 

M& =— Q \ h xl(<Pu - s <Pix) + + ((fit - W lx )~]dA + T 

(9.1.22) 

M iyi = —2Qg\ (yi+xOzi)hdx—Q\ h y [((p lt -s <p lx ) + +((p lt —s (p lx y]dA 

(9.1.23) 



hi0 31 =-2Qgd 31 i (y-y' c )hdA-2 Q g yi 



xhdx 

L 



-2^31 j < 



x 2 hdx+lT 



-q\ [xh y -(y-y'jh x ][(<p lt -s (p lx )+ + ((p lt -s (p lx )-]dJ. 

We note that integrals over the projected water-line L of the ship on 
the vertical plane when in its equilibrium position occur in addition 
to integrals over the vertical projection A of the entire hull. The 
quantity 7 31 arises from the relation I = fil 31 for the moment of 
inertia / of the ship with respect to an axis through its e.g. parallel 
to the z'-axis. The equation (9.1.21) determines the surge s l9 and also 
the speed s (or. if one wishes, the thrust T is determined if s is 



256 WATER WAVES 

assumed to be given). Furthermore, the speed s is fixed solely by T 
and the geometry of the ship's hull. This can be seen, with reference 
to (9.1.14) and the discussion that accompanies it, in the following 
way: The term y) (x, y, z) in (9.1.14) is the term in <p x that is indepen- 
dent of t. It therefore determines T upon insertion of q? 1 in (9.1.21). 
This term, however, is obtained by finding the harmonic function 
ip as a solution of the boundary problem for ip formulated in (9.1.14 ) . 
In fact, the relation between s and T is now seen to be exactly the 
same relation as was obtained by Michell. (It will be written down in 
a later section. ) In other words, the wave resistance depends only on 
the basic translatory motion with uniform speed of the ship, and not 
at all on its small oscillations relative to that motion. If, then, effects 
on the wave resistance due to the oscillation of the ship are to be 
obtained from the theory, it will be necessary to take account of higher 
order terms. Once the thrust T has been determined the equations 
(9.1.22) and (9.1.23) form a coupled system for the determination of 
y 1 and 3V since (p 1 and d n have presumably been determined previous- 
ly. Thus our system is one in which there is a considerable amount of 
cross-coupling. It might also be noted that the trim, i.e. the constant 
values of y 1 and 31 about which the oscillations in these degrees of 
freedom occur are determined from (9.1.22) and (9.1.23) by the time- 
independent terms in these equations — including, for example, the 
moment IT of the thrust about the e.g. 

We proceed to the discussion of other conclusions arising from our 
developments and concerning two questions which recur again and 
again in the literature. These issues center around the question: what 
is the correct manner of satisfying the boundary conditions on the 
curved hull of the ship? Michell employed the condition (9.1.12), 
naturally with 6 n = 21 = co 1 = 0, on the basis of the physical argu- 
ment that s h x represents the component of the velocity of the water 
normal to the hull, and since the hull is slender, a good approximation 
would result by using as boundary condition the jump condition 
furnished by (9.1.12). Havelock and others have usually followed the 
same practice. However, one finds constant criticism of the resulting 
theory in the literature (particularly in the engineering literature) 
because of the fact that the boundary condition is not satisfied at the 
actual position of the ship's hull, and various proposals have been 
made to improve the approximation. This criticism would seem 
to be beside the point, since the condition (9.1.12) is simply the con- 
sequence of a reasonable linearization of the problem. To take account 



THE MOTION OF A SHIP IN A SEAWAY 



257 



of the boundary condition at the actual position of the hull would, of 
course, be more accurate — but then, it would be necessary to deal 
with the full nonlinear problem and make sure that all of the essential 
correction terms of a given order were obtained. In particular, it 
would be necessary to examine the higher order terms in the condi- 
tions at the free surface — after all, the conditions (9.1.13), which are 
also used by Michell and Havelock (and everyone else, for that 
matter), are satisfied at y = and not on the actual displaced position 
of the free surface. One way to obtain a more accurate theory would 
be, of course, to carry out the perturbation scheme outlined here to 
higher order terms. 

Still another point has come up for frequent discussion (cf., for 
example, Lunde and Wigley [L.18]) with reference to the boundary 
condition on the hull. It is fairly common in the literature to refer to 
ships of Michell's type, by which is meant ships which are slender 
not only in the fore-and-aft direction, but which are also slender in 
the cross-sections at right angles to this direction (cf. Fig. 9.1.3) so 

1 y 




(a) (b) 

Fig. 9.1.3a, b. Ships with full and with narrow mid- sections 



that h y , in our notation, is small. Thus ships with a rather broad 
bottom (cf. Fig. 9.1.3a), or, as it is also put, with a full mid-section, 
are often considered as ships to which the present theory does not 
apply. On the other hand, there are experimental results (cf. Havelock 
[H.7]) which indicate that the theory is just as accurate for ships 
with a full mid-section as it is for ships of Michell's type. When the 
problem is examined from the point of view taken here, i.e. as a 
problem to be solved by a development with respect to a parameter 
characterizing the slenderness of the ship, the difference in the two 
cases would seem to be that ships with a full mid-section should be 
regarded as slender in both draft and beam, (otherwise no lineariza- 
tion based on assuming small disturbances in the water would be 



258 WATER WAVES 

reasonable), while a ship of Michell's type is one in which the draft is 
finite and the beam is small. In the former case a development dif- 
ferent from the one given above would result: the mass and moments 
of inertia would be of second order, for instance, rather than first 
order. Later on we shall have occasion to mention other possible ways 
of introducing the development parameter. 

We continue by pointing out a number of conclusions, in addition 
to those already given, which can be inferred from our equations 
without solving them. Consider, for example, the equations (9.1.22) 
and (9.1.23) for the heave y 1 and the pitching oscillation 31 , and make 
the assumption that 

(9.1.24) f xhdx = 

(which means that the horizontal section of the ship at the water line 
has the e.g. of its area on the same vertical as that of the whole ship). 
If this condition is satisfied it is immediately seen that the oscillations 
31 and y 1 are not coupled. Furthermore, these equations are seen to 
have the form 

(9.1.25) y t -\-?*y 1 =p(t) 

(9.1.26) d3i+*p3i = q(t) 
with 

(9.1.27) X\ = 

(9.1.28) X\ = 



2Qg j L hdx 

2£g [\ A (y~ V>dA + J l x*hdx~\ 



It follows that resonance* is possible if p(t) has a harmonic component 
of the form A cos (X x t -f- B) or q(t) a component of the form 
A cos (X 2 t + B): in other words, one could expect exceptionally 
heavy oscillations if the speed of the ship and the seaway were to be 
such as to lead to forced oscillations having frequencies close to these 
values. One observes also that these resonant frequencies can be 
computed without reference to the motion of the sea or the ship: 
the quantities X r , A 2 depend only on the shape of the hull.** 

* The term resonance is used here in the strict sense, i.e. that an infinite 
amplitude is theoretically possible at the resonant frequency. 

** The equation (9.1.27) can be interpreted in the following way: it furnishes 
the frequency of free vibration of a system with one degree of freedom in which 
the restoring force is proportional to the weight of water displaced by a cylinder 
of cross-section area 2 [ L hdx when it is immersed vertically in water to a depth y x . 



THE MOTION OF A SHIP IN A SEAWAY 259 

In spite of the fact that the linear theory presented here must be 
used with caution in relation to the actual practical problems con- 
cerning ships in motion, it still seems likely that such resonant fre- 
quencies would be significant if they happened to occur as harmonic 
components in the terms p(t) or q(t) with appreciable amplitudes. 
Suppose, for instance, that the ship is moving in a sea-way that 
consists of a single train of simple harmonic progressing plane waves 
with circular frequency a which have their crests at right angles to 
the course of the ship. If the speed of the ship is s one finds that the 
circular excitation frequency of the disturbances caused by such 
waves, as viewed from the moving coordinate system (cc, y, z) that is 
used in the discussion here, is a + s a 2 /g, since a 2 /g is 2n times the 
reciprocal of the wave length of the wave train. Thus if X x or X 2 should 
happen to lie near this value, a heavy oscillation might be expected. 
One can also see that a change of course to one quartering the waves at 
angle y would lead to a circular excitation frequency a -\-s cos y • o 2 /g 
and naturally this would have an effect on the amplitude of the response. 

It has already been stated that the theory presented here is closely 
related to the theory published by Haskind [H.4] in 1946, and it was 
indicated that the two theories differ in some respects. We have not 
made a comparison of the two theories in the general case, which would 
not be easy to do, but it is possible to make a comparison rather easily 
in the special case treated by Haskind in detail. This is the special 
case dealt with in the second of his two papers in which the ship is 
assumed to oscillate only in the vertical plane — as would be possible 
if the seaway consisted of trains of plane waves all having their crests 
at right angles to the course of the ship. Thus only the quantities y x (t) 
and 31 (t), which are denoted in Haskind's paper by f (t) and \p(t ), are of 
interest. Haskind finds differential equations of second order for these 
quantities, but these equations are not the same as the corresponding 
equations (9.1.22), (9.1.23) above. One observes that (9.1.22) con- 
tains as its only derivative the second derivative^ and (9.1.23) con- 
tains as its sole derivative a term with 31 ; in other words there are no 
first derivative terms at all, and the coupling arises solely through 
the undifferentiated terms. Haskind's equations are quite different 
since first and second derivatives of both dependent functions occur 
in both of the two equations; thus Haskind, on the basis of his theory, 
can speak, for example, of damping terms, while the theory presented 
here yields no such terms. On the basis of the theory presented so far 
there should be no damping terms of this order for the following 



260 WATER WAVES 

reasons: In the absence of frictional resistances, the only way in 
which energy can be dissipated is through the transport of energy to 
infinity by means of out-going progressing waves. However, we have 
already given valid reasons for the fact that those oscillations of the 
ship which consist solely of displacements parallel to the vertical 
plane produce waves in the water with amplitudes that are of higher 
order than those considered in the first approximation. Thus no such 
dissipation of energy should occur.* In any case, our theory has this 
fact as one of its consequences. Haskind [H.4] also says, and we quote 
from the translation of his paper (see page 59): "Thus, for a ship 
symmetric with respect to its midship section . . ., only in the absence 
of translatory motion, i.e., for s = 0, are the heaving and pitching 
oscillations independent." This statement does not hold in our version 
of the theory. As one sees from (9.1.22) and (9.1.23) coupling occurs 

if, and only if, xhdx =£ 0, whether s vanishes or not. In addition, 



Haskind obtains no resonant frequencies in these displacements be- 
cause of the presence of first-derivative terms in his equation; the 
author feels that such resonant frequencies may well be an important 
feature of the problem. Thus it seems likely that Haskind's theory 
differs from that presented here because he includes a number of 
terms which are of higher order than those retained here. Of course, it 
does not matter too much if some terms of higher order are included 
in a perturbation theory, at least if all the terms of lowest order are 
really present: at worst, one might be deceived in giving too much 
significance to such higher order terms. 

The fact that the theory presented so far leads to the conclusion 
that no damping of the pitching, surging, and heaving oscillations 
occurs is naturally an important fact in relation to the practical pro- 
blems. Unfortunately, actual hulls of ships seem in many cases to be 
designed in such a way that damping terms in the heaving and pitch- 
ing oscillations are numerically of the same order as other terms in 
the equations of motion of a ship. (At least, there seems to be experi- 
mental evidence from model studies — see the paper by Korvin- 
Krukovsky and Lewis [K. 16]— which bears out this statement.) 
Consequently, one must conclude that either actual ships are not 

* It is, however, important to state explicitly that there would be damping 
of the rolling, yawing, and swaying oscillations, since these motions create waves 
having amplitudes of the order retained in the first approximation, and thus 
energy would be carried off to infinity as a consequence of such motions. 



THE MOTION OF A SHIP IN A SEAWAY 261 

sufficiently slender for the lowest order theory developed here to apply 
with accuracy, or that important physical factors such as turbulence, 
viscosity, etc., have effects so large that they cannot be safely neg- 
lected. If it is the second factor that is decisive, rather than the loss 
of energy due to the creation of waves through pitching and heaving, 
it is clear that only a basic theory different from the one proposed 
here would serve to include such effects. If, however, the damping 
has its origin in the creation of gravity waves we need not be entirely 
helpless in dealing with it in terms of the sort of theory contemplated 
here. It would not be helpful, though, to try to overcome the difficulty 
by carrying the development to terms of higher order, for example, 
even though there would certainly then be damping effects in pitching 
and heaving: such damping effects of higher order could evidently not 
introduce damping into the motions of lower order. This is fortunately 
not the only way in which the difficulty can be attacked. One rather 
obvious procedure would be to retain the present theory, and simply 
add damping terms with coefficients to be fixed empirically, in some- 
what the same fashion as has been proposed by St. Denis and Wein- 
blum [S.l], for example. 

There are still other possibilities for the derivation of theories which 
would include damping effects without requiring a semi-empirical 
treatment, but rather a different development with respect to a slen- 
derness parameter. One such possibility has already been hinted at 
above in the course of the discussion of ships of broad mid-section 
compared with ships of Michell's type. If the ship is considered to be 
slender in both draft and beam the waves due to oscillations of the 
ship would be of the same order with respect to all of the degrees of 
freedom; a theory utilizing this observation is being investigated. 
Another possibility would be to regard the draft as small while the 
beam is finite (thus the ship is thought of as a flat body with a 
planing motion over the water), i.e. to base the perturbation scheme 
on the following equation for the hull (instead of (9.1.2)): 

and to carry out the development with respect to /?. This theory has 
been worked out in all detail, though it has not yet been published. 
With respect to damping effects the situation is now in some respects 
just the reverse of that described above: now it is the oscillations in 
the vertical plane, together with the rolling oscillation, that are 
damped to lowest order, while the yawing and swaying oscillations 



262 



WATER WAVES 



are undamped. It would seem reasonable therefore to investigate the 
results of such a theory for conventional hulls and make comparisons 
with model experiments. This still does not exhaust all of the possibili- 
ties with respect to various types of perturbation schemes, particu- 
larly if hulls of special shape are introduced. Consider, for example, 
a hull of the kind used for some types of sailing yachts, and shown 
schematically in Fig. 9.1.4. Such a hull has the property that its beam 




Fig. 9.1.4. Cross section of hull of a yacht 



and draft are both finite, but the hull cross section consists of two 
thin disks joined at right angles like a T. In this case an appropriate 
development with respect to a slenderness parameter can also be 
made in regarding both disks as being slender of the same order. The 
result is a theory in which all oscillations, except the surge, would be 
damped; this theory has been worked out too but not yet published. 
It would take up an inordinate amount of space in this book to deal 
in detail with all of the various types of possible perturbation schemes 
mentioned above. In addition, only one of them seems so far to permit 
explicit solutions even in special cases, and that is the generalization 
of the Michell theory which was explained at some length above. Con- 
sequently, only this theory (in fact, only a special case of it) will be 
developed in detail in the remainder of the chapter. In all other 
theories, it seems necessary to solve certain integral equations before 
the motion of the ship can be determined even under the most restric- 
tive hypotheses — such as a motion of pure translation with no oscilla- 
tions whatever, for example. Even in the case of the generalized 
Michell theory (i.e. the case of a ship regarded as a thin disk disposed 
vertically) an explicit solution of the problem for the lowest order 
approximation (p ± to the velocity potential — in terms of an integral 



THE MOTION OF A SHIP IN A SEAWAY 263 

representation, say— seems out of the question. In fact, as soon as 
rolling or yawing motions occur, explicit solutions are unlikely to be 
found. The best that has been done so far in such cases has been to 
formulate an integral equation for the values of (p 1 over the vertical 
projection A of the ship's hull; this method of attack, which looks 
possible and somewhat hopeful for numerical purposes since the 
motion of the ship requires the knowledge of <p ± only over the area A, 
is under investigation. However, if the motion of the ship is confined 
to a vertical plane, so that co 1 = 6 n = d 21 = 0, it is possible to solve 
the problems explicitly. This can be seen with reference to the bound- 
ary conditions (9.1.12) and (9.1.13) which in this case are identical 
with those of the classical theory of Michell and Havelock, and hence 
permit an explicit solution for <p 1 which is given later on in section 
9.4. After 9? x is determined, it can be inserted in (9.1.21), (9.1.22), and 
(9.1.23) to find the forward speed s corresponding to the thrust T, 
the two quantities fixing the trim, and the surge, pitching, and heav- 
ing oscillations.* In all, six quantities fixing the motion of the ship 
can be determined explicitly. Only this version of the theory will be 
presented in detail in the remainder of the chapter. 

The theory discussed here is very general, and it therefore could be 
applied to the study of a wide variety of different problems. For exam- 
ple, the stability of the oscillations of a ship could be in principle 
investigated on a rational dynamical basis, rather than as at present 
by assuming the water to remain at rest when the ship oscillates. It 
would be possible to investigate theoretically how a ship would move 
with a given rudder setting, and find the turning radius, angle of heel, 
etc. The problem of stabilization of a ship by gyroscopes or other de- 
vices could be attacked in a very general way: the dynamical equa- 
tions for the stabilizers would simply be included in the formulation 
of the problem together with the forces arising from the interactions 
of the water with the hull of the ship. 

In sec. 9.2 the general formulation of the problem is given; in 
sec. 9.3 the details of the linearization process are carried out for the 
case of a ship which is slender in beam (i.e. under the condition 
implied in the classical Michell-Havelock theory); and in sec. 9.4 a 
solution of the problem is given for the case of motion confined to the 
vertical plane, including a verification of the fact that the wave 
resistance is given by the same formula as was found by Michell. 

* These free undamped vibrations are uniquely determined only when initial 
conditions are given. 



264 



WATER WAVES 



9.2. General formulation of the problem 

We derive here a theory for the most general motion of a rigid body 
through water of infinite depth which is in its turn also in motion in 
any manner. As always we assume that a velocity potential exists. 
Since we deal with a moving rigid body it is convenient to refer the 
motion to various types of moving coordinate systems as well as to a 
fixed coordinate system. The fixed coordinate system is denoted by 
O — X, Y, Z and has the disposition used throughout this book: The 
X, Z-plane is in the equilibrium position of the free surface of the 
water, and the F-axis is positive upwards. The first of the two moving 
coordinate systems we use (the second will be introduced later) is 
denoted by o — x,y,z and is specified as follows (cf. Fig. 9.2.1): 




Fig. 9.2.1. Fixed and moving coordinate system 

The cc, 2-plane coincides with the X, Z-plane (i.e. it lies in the undis- 
turbed free surface), the y-axis is vertically upward and contains the 
center of gravity of the ship. The #-axis has always the direction of the 
horizontal component of the velocity of the center of gravity of the 
ship. (If we define the course of the ship as the vertical projection of 
the path of its center of gravity on the X, Z-plane, then our conven- 
tion about the ^-axis means that this axis is taken tangent to the 
ship's course.) Thus if R c = (X c , Y c , Z c ) is the position vector of the 
center of gravity of the ship relative to the fixed coordinate system 
and hence R c = (X c , Y c , Z c ) is the velocity of the e.g., it follows that 
the a?-axis has the direction of the vector u given by 

(9.2.1) u = X c I + Z c K 

with I and K unit vectors along the X-axis and the Z-axis. If i is a 
unit vector along the ^-axis we may write 

(9.2.2) s(t)i = u, 



THE MOTION OF A SHIP IN A SEAWAY 265 

thus introducing the speed s(t) of the ship relative to a horizontal 
plane. For later purposes we also introduce the angular velocity 
vector co of the moving coordinate system: 

(9.2.3) co = co(t)J, 
and the angle a (cf. Fig. 9.2.1) by 

(9.2.4) a(/) = oj(T)dr. 

J o 

The equations of transformation from one coordinate system to the 
other are 

(X=x cos oc+2 sin 0L J r X c ; x=(X — X c ) cos a — (Z — Z c ) sin a 
Y=y ;y = Y 

Z=— x sin tx-\-z cos a. -\-Z c ;z = (X— X c ) sin a + (Z— Z c )cosa. 
By 0{X, Y, Z; t) we denote the velocity potential and write 
(9.2.6) 0(X, Y, Z; t) 

= 0(x cos a + z sin a + X c , y, — x sin a + z cos a + Z c \ t) 
= <p(x, y, z; t). 

In addition to the transformation formulas for the coordinates, we 
also need the formulas for the transformation of various derivatives. 
One finds without difficulty the following formulas: 

{0 X = <p x cos « + <Pz sm a 
@Z = — <Px Sm a + <Pz COS a - 

It is clear that grad 2 0(X, Y, Z; t) = grad 2 qp(x, y, z; t) and that rp is 
a harmonic function in x, y, z since is harmonic in X, Y, Z. To cal- 
culate t is a little more difficult; the result is 

(9.2.8) t =-( 8 + ojz)cp x + coxcp z + (p t . 

(To verify this formula, one uses t = (p x x t + cp y y t + <p z z t + (ft an d 
the relations (9.2.5) together with s cos a = X c , s sin a = — Z c -) The 
last two sets of equations make it possible to express Bernoulli's law 
in terms of cp(x, y, z; t); one has: 

(9.2.9) V- + gy + — (grad 9p) 2 - (* + ojz)<p x + oio^, + (p t = 0. 
P 2 

Suppose now that F(X, F, Z; Z) = is a boundary surface (fixed 
or moving) and set 

(9.2.10) F(x cos a -f . . ., y, — x sin a + . • •; t) = /(a?, ?/, 2; /), 



266 



WATER WAVES 



so that f(x, y, z; t) = is the equation of the boundary surface rela- 
tive to the moving coordinate system. The kinematic condition to be 
satisfied on such a boundary surface is that the particle derivative 
dF/dt vanishes, and this leads to the boundary condition 

(9.2.11) cp x f x + (p y f y + cp z f z - (s + (oz)f x + coxf z +f t = 

relative to the moving coordinate system upon using the appropriate 
transformation formulas. In particular, if y — r)(x, z; t) = is the 
equation of the free surface of the water, the appropriate kinematic 
condition is 

(9.2.12) - (p x t] x + cp y - (p z tj z + (s + ojz)rj x - a>xr\ z — rj t = 

to be satisfied for y — r\. (The dynamic free surface condition is of 
course obtained for y = r\ from (9.2.9) by setting p = 0.) 

We turn next to the derivation of the appropriate conditions, both 
kinematic and dynamic, on the ship's hull. To this end it is convenient 
to introduce another moving coordinate system o' — x' , y' , z' which 
is rigidly attached to the ship. It is assumed that the hull of the ship 
has a vertical plane of symmetry (which also contains the center of 
gravity of the ship); we locate the x', ?/'-plane in it (cf. Fig. 9.2.2) and 
suppose that the y'-axis contains the center of gravity. The o' — x\ 
y\ z' system, like the other moving system, is supposed to coincide 



*y 




^m^/////////^ 



(a) 



(b) 



Fig. 9.2.2a, b. Another moving coordinate system 



with the fixed system in the rest position of equilibrium. The center 
of gravity of the ship will thus be located at a definite point on the 
?/'-axis, say at distance y' c from the origin o' : in other words, the system 
of coordinates attached rigidly to the ship is such that the center of 
gravity has the coordinate (0, y' c , 0). 

In the present section we do not wish in general to carry out lineari- 
zations. However, since we shall in the end deal only with motions 



THE MOTION OF A SHIP IN A SEAWAY 267 

which involve small oscillations of the ship relative to the first moving 
coordinate system o — x, y, z, it is convenient and saves time and 
space to suppose even at this point that the angular displacement 
of the ship relative to the o — x, y, z system is so small that it can 
be treated as a vector 8: 

(9.2.13) 6=^1+ OJ + 3 k. 

The transformation formulas, correct up to first order terms in the 
components d { of 8, are then given by: 

(x'=x+ 6 3 (y - y c ) - d 2 z 

(9.2.14) \y' = y-{y c - y' c ) + o 1 z - e 3 x 

[z' = z +d 2 x - e x (y - y c ) 

with y c of course representing the ^-coordinate of the center of 
gravity in the unprimed system. It is assumed that y c — y' e is a small 
quantity of the same order as Q t and only linear terms in this quantity 
have been retained. (The verification of (9.2.14) is easily carried out 
by making use of the vector-product formula 8 = 8 X r, for the 
small displacement 8 of a rigid body under a small rotation 8.) 

The equation of the hull of the ship (assumed to be symmetrical 
with respect to the x', ?/-plane) is now supposed given relative to the 
primed system of coordinates in the form: 

(9.2.15) z' = ±£(x',y'), z'%0. 

The equation of the hull relative to the o — x, y, z-system can now 
be written in the form 

(9.2.16) z + 0^-d^y - y' c ) - £(x, y) + [6 2 z - d 3 (y - y' c )]£ x (x,y) 

+ i(y c - y' c ) - o lZ + d s x]C y (x, y) = 0, *' > o, 

when higher order terms in (y c — y' c ) and ^ are neglected. The left 
hand side of this equation could now be inserted for / in (9.2.11) to 
yield the kinematic boundary condition on the hull of the ship, but 
we postpone this step until the next section. 

The dynamical conditions on the ship's hull are obtained from the 
assumption that the ship is a rigid body in motion under the action 
of the propeller thrust T, its weight Mg, and the pressure p of the 
water on its hull. The principle of the motion of the center of gravity 
yields the condition 

(9.2.17) M ~ (si +yj) = [ pn dS + T - Mgj. 



268 WATER WAVES 

By n we mean the inward unit normal on the hull. Our moving 
coordinate system o — x, y, z is such that di/dt = — cok and d]jdt =■ 
0, so that (9.2.17) can be written in the form 



(9.2.18) M'si — Mswk + My j 



pndS + T- Mgl 



J s 



with p defined by (9.2.9). The law of conservation of angular momen- 
tum is taken in the form: 

(9.2.19) i.[ (R - R c ) X (R - R c )dm 
at J M 



J 



p(R - R c ) n dS + (R T - R c ) X T. 

s 

The crosses all indicate vector products. By R is meant the position 
vector of the element of mass din relative to the fixed coordinate 
system. R c (cf. Fig. 9.2.1) fixes the position of the e.g. and R r 
locates the point of application of the propeller thrust T. We introduce 
r = (x, y, z) as the position vector of any point in the ship in the 
moving coordinate system and set 

(9.2.20) q = r - yj, 

so that q is a vector from the e.g. to any point in the ship. The relation 

(9.2.21) R = R c + (co +6) X q 

holds, since co + 6 is the angular velocity of the ship; thus (9.2.21) 
is simply the statement of a basic kinematic property of rigid bodies. 
By using the last two relations the dynamical condition (9.2.19) can 
be expressed in terms of quantities measured with respect to the 
moving coordinate system o — x, y, z, as follows: 

(9.2.22) - f (r - yj) X [(co + 6) X (r - yj)]dm 
dtJ M 

p(r-yj) xndS + (R T -R c )xT. 



1 



We have now derived the basic equations for the motion of the 
ship. What would be wanted in general would be a velocity potential 
(p(x, y, z; t) satisfying (9.2.11) on the hull of the ship, conditions 
(9.2.9) (with p = 0) and (9.2.12) on the free surface of the water; 
and conditions (9.2.17) and (9.2.22), which involve op under integral 
signs through the pressure p as given by (9.2.9). Of course, the quan- 



THE MOTION OF A SHIP IN A SEAWAY 269 

tities fixing the motion of the ship must also be determined in such a 
way that all of the conditions are satisfied. In addition, there would 
be initial conditions and conditions at oo to be satisfied. Detailed 
consideration of these conditions, and the complete formulation of the 
problem of determining 99 (x, y, z; t) under various conditions will be 
postponed until later on since we wish to carry out a linearization 
of all of the conditions formulated here. 



9.3. Linearization by a formal perturbation procedure 

Because of the complicated nature of our conditions, it seems wise 
(as was indicated in sec. 1 of this chapter) to carry out the lineari- 
zation by a formal development in order to make sure that all terms of 
a given order are retained; this is all the more necessary since terms 
of different orders must be considered. The linearization carried out 
here is based on the assumption that the motion of the water relative 
to the fixed coordinate system is a small oscillation about the rest 
position of equilibrium. It follows, in particular, that the elevation of 
the free surface of the water should be assumed to be small and, of 
course, that the motion of the ship relative to the first moving coor- 
dinate system — cc, y, z should be treated as a small oscillation. We 
do not, however, wish to consider the speed of the ship with respect 
to the fixed coordinate system to be a small quantity: it should rather 
be considered a finite quantity. This brings with it the necessity to 
restrict the form of the ship so that its motion through the water does 
not cause disturbances so large as to violate our basic assumption; 
in other words, we must assume the ship to have the form of a thin 
disk. In addition, it is clear that the velocity of such a disk-like ship 
must of necessity maintain a direction that does not depart too much 
from the plane of the thin disk if small disturbances only are to be 
created. Thus we assume that the equation of the ship's hull is given by 

(9.3.1) z' = /3h(x\y'), z' > 0, 

with /? a small dimensionless parameter, so that the ship is a thin 
disk symmetrical with respect to the x', ?/'-plane, and (5h takes the 
place of f in (9.2.15). (It has already been noted in the introduction 
to this chapter that this is not the most general way to describe the 
shape of a disk that would be suitable for a linearization of the type 
carried out here.) We have already assumed that the motion of the 
ship is a small oscillation relative to the moving coordinate system 



(9.3.2) 


<p(x,y 


(9.3.3) 


T}{x 


(9.3.4) 




(9.3.5) 




(9.3.6) 




(9.3.7) 





270 WATER WAVES 

o — x, y, z. It seems reasonable, therefore, to develop all our basic 
quantities (taken as functions of x, y, z; t) in powers of /3, as follows: 

, z; t; P) = pep, + p\ 2 + . . ., 

, z; t; P) = p Vl + Pn % + . . ., 
s(t; P) = s + Ps x + P*s 2 + . . ., 
co(£; P) = a) + /fc^ + /? 2 co 2 + • • -, 
d i (t;P)=pS il + / W« + ..., t = l f 2,8 

y c - y' c = fyi + ^ 2 «/ 2 + • • •• 

The first and second conditions state that the velocity potential and 
the surface wave amplitudes, as seen from the moving system, are 
small of order p. The speed of the ship, on the other hand, and the 
angular velocity of the moving coordinate system about the vertical 
axis of the fixed coordinate system, are assumed to be of order zero. 
(It will turn out, however, that co must vanish — a not unexpected 
result.) The relations (9.3.6) and (9.3.7) serve to make precise our 
assumption that the motion of the ship is a small oscillation relative 
to the system o — x, y, z. 

We must now insert these developments in the conditions derived 
in the previous section. The free surface conditions are treated first. 
As a preliminary step we observe that 

(9.3.8) (p x (z, rj, z; t; P) = P[(p lx (x, 0, z; t) + r)(p lxy (x, 0, z; t) + . . .] 

+ P 2 [<P2X +W2XV + • • •] 

+ 

= P<p lx (x, 0, z; t) + a [i7i9W(a?, °> z "> + <P2x(®> °> z > 01 

+ , 

with similar formulas for other quantities when they are evaluated 
on the free surface y = r\. Here we have used the fact that rj is small 
of order p and have developed in Taylor series. Consequently, the 
dynamic free surface condition for y = rj arising from (9.2.9) with 
p ■= can be expressed in the form 

(9.3.9) g[p Vl +p*rj 2 + . .-.]+i^[(grad^)» + . . .] 

- [*o + 0*1 +; • • + 2 (^o + A»! + • • .)]fc + 

P*(ni<Pi*v + ^x) + • • •] 

+x(o) Q + Pa h + . . .)[P<p u + P 2 (r)mzy + ^J + • • •] 

+ [fait +P 2 (Vl<Plty +<P*t) +'••] =° 

and this condition is to be satisfied for y = 0. In fact, as always in 



THE MOTION OF A SHIP IN A SEAWAY 271 

problems of small oscillations of continuous media, the boundary 
conditions are satisfied in general at the equilibrium position of the 
boundaries. Upon equating the coefficient of the lowest order term 
to zero we obtain the dynamical free surface condition 

(9.3.10) - gr\ x + (s + a)<p)<p lx - w x<p lz — <p lt = for y = 0, 

and it is clear that conditions on the higher order terms could also be 
obtained if desired. In a similar fashion the kinematic free surface 
condition can be derived from (9.2.12); the lowest order term in f$ 
yields this condition in the form: 

(9.3.11) <p lv + (s + co^)rj lx - co xt] lz - rj lt = for y = 0. 

We turn next to the derivation of the linearized boundary condi- 
tions on the ship's hull. In view of (9.3.6) and (9.3.7), the transforma- 
tion formulas (9.2.14) can be put in the form 

fx' = x + po n (y - y' c ) - 00 21 s 

(9.3.12) \v' = v-pyi+ 00ii* - 0031* 

when terms involving second and higher powers of /9 are rejected. 
Consequently, the equation (9.2.16) of the ship's hull, up to terms in 
2 , can be written as follows: 

z + pe n x - pe xl (y - y' c ) - ?h[x + pd n (y - y' c ) - /3d 21 z, 

y-fyi+ 00ii* - 0031*] = o, 

and, upon expanding the function h, the equation becomes 

(9.3.13) z + 06 21 x - 00 n (t/ - y' c ) - ph(x, y) + . . . = 0, 

the dots representing higher order terms in p. We can now obtain the 
kinematic boundary condition for the hull by inserting the left hand 
side of (9.3.13) for the function / in (9.2.11); the result is 

co = 



(9.3.14; 

<Pu = *o(0 2 i - K) - m»i - 02i* + dn(y - y c ) 

when the terms of zero and first order only are taken into account. 
It is clear that these conditions are to be satisfied over the domain 
A of the x, ?/-plane that is covered by the projection of the hull on the 
plane when the ship is in the rest position of equilibrium. As was 
mentioned earlier, it turns out that oj = 0, i.e., that the angular 
velocity about the y-axis must be small of first order, or, as it could 
also be put, the curvature of the ship's course must be small since the 



272 WATER WAVES 

speed in the course is finite. The quantity s ± (t) in (9.3.4) evidently 
yields the oscillation of the ship in the direction of the «-axis (the so- 
called "surge"). 

It should also be noted that if we use z' = — {3h(x', y') we find, 
corresponding to (9.3.14), that 

<Piz = s (0 21 + h x ) - K + Q 21 )x + n (y - y' c ). 
This means that A must be regarded as two sided, and that the last 
equation is to be satisfied on the side of A which faces the negative 
z-axis. The last equation and (9.3.14) imply that (p may have discon- 
tinuities at the disk A. 

The next step in the procedure is to substitute the developments 
with respect to /?, (9.3.2) — (9.3.7), in the conditions for the ship's 
hull given by (9.2.18) and (9.2.22). Let us begin with the integral 



I 



pn dS which appears in (9.2.18). In this integral S is the immersed 



surface of the hull, n is the inward Unit normal to this surface and p 
is the pressure on it which is to be calculated from (9.2.9). With 
respect to the o — x , y, z coordinate system the last equations of the 
symmetrical halves of the hull are 

/o S1 ^ i s i' z = H 1 (x,y;t;P) = f x + / 2 

K • ' } \S 2 : z = H 2 (x, y; t; 0) = - f x + / 2 

where 

/q i i«i I /i = ^ + ^[ 3i(2/ - y' c )K - (0 n x + yi )h v ] + o(P) 

(9.d.lb) \ h= _ ^ Kx + P6ii{y _ ylj + Q(n 

We can now write 

pn dS = pn-t dS 1 + pn 2 dS 2 
Js Js t Js 2 

in which n ± and n 2 are given by 

H lx i + II ly j - k - H 2x i - H 2 J + k 



Vl + #?,+#*/ Vl+Hl+H*,' 

We can also write 

pn dS = — ggi yndS + \ Pl n dS 

Js Js Js 

= - Qg\ yndS + \ p 1 n 1 dS 1 + p x n 2 dS, 

JS JSx *'S 2 

where p l9 from (9.2.9), is given by 

(9.3.17) p 1 = — ^[J(grad (p) 2 — (s 4- wz)<p x + xco(p z + <p t ]. 



THE MOTION OF A SHIP IN A SEAWAY 



273 



If S is the hull surface below the x, 2-plane, the surface area S — S 
is of order @ and in this area each of the quantities y, H v H 2 is of 
order p. Hence one finds the following to hold: 

- f yn dS = - f y n dS + (i + j)O(^) + kOtf*). 

Js Js 

From the divergence theorem we have 

yn dS = VI 

where V is the volume bounded by S and the x, z-plane. With an 
accuracy of order /? 3 , V is given by 



*L 



MA 



P(y x +d zl x)dB=2p 



hdA—2jP\ (y^d^hdx. 



Here A is the projection of the hull on the vertical plane when the 
hull is in the equilibrium position, B is the equilibrium water line 
area, and L is the projection of the equilibrium water line on the 
.z-axis. 

If W l9 W 2 are the respective projections of the immersed surfaces 
S l3 S 2 on the x, y-plane we have 



p ± ndS = i p x (x, y, H t ; t)H lx dW r 

J s \Jw 1 

Pl (x, y, H i; t)H ly dW r 



-k 



p x (x, y, H 2 ; t)H 2x dW^ 



p x (x, y, H 2 ; t)H 2v dWc t 



p x (x, y, H i; t)dW 1 — p x (x, y, H 2 ; t)dW 2 
w x J w 2 

Neither W x nor W 2 is identical with A. Each of the differences 
W t — A, W 2 — A is, however, an area of order /?. From this and the 
fact that each of the quantities p, H lx , H ly , H 2x , H 2y is of order /?, it 
follows that 



(9.3.18) 



Pl n dS=ilj [ Vl {x, y, H t ; t)H lx - Pl (x, y, H 2 ;t)H 2x ]dA+0(p*) 
+J { f [Pi(x, V, H i; t)H ly - Pl (x, y, H 2 ; t)H 2y ]dA +0(/? 3 ) 
[ Pl (x, y, H i; t)- Pl (x, y, H 2 ; t)}dA +0(^)\ . 



274 



WATER WAVES 



It was pointed out above that 99 may be discontinuous on A. Hence 
from (9.3.17), (9.3.2), (9.3.4) we write 



(9.3.19 



Pl (x, y, H i; t) = Q p(s cp lx - <p lt ) + + 0(P*] 
Pi(x, y, H 2 ; t) = gP{s (p lx - tp lt )- + 0(fi 2 \ 



(9.3.20) +j 



Here the + and — superscripts denote values at the positive and 
negative sides of the disk A whose positive side is regarded as the side 
which faces the positive z-axis. If we substitute the developments of 
H lx , H ly , H 2x , H 2y , and (9.3.19) in (9.3.18), then collect the previous 
results, we find 

f pn dS=i J Q p f [(^-0 21 )( W i,-^) + + (^+^2i)(Wi,-^)-]^+^ 3 ) J 

2Qgpt hdA-2Qgp*( ( yi +xd 31 )hdx 

+eP*j [(^+0ii)(Wi»-^i*) + +(^-^ii)(Wix-^)-]^+O(^) 

~ k {^f l( s o9i,-^it) + -(s f lx -cp lt )-]dA^O(P)i 

The integral p(r — y c j) X n dS which appears in (9.2.22) can 
be written 

P(r-y c j)XndS=-Qg\ y(r-y c j)xndS 

+ Pi(r-y < i)Xn 1 dS l 
Js 1 



+ Pi(r-yJ)xn 2 dS t . 

J s 



If we use the same procedure as was used above for the expansion of 
pn dS we find 



J> 



r—y c j)xndS=-i 



[ IqP [(y-yc)(so<Pix-Vit) + -(y-yc)(soVix-<pu)-]dA+o(py 

+j QP [v(s»<Pix —<Pit) + ~ *(Wiaf - <Pu)~]dA +0(£ 2 ) 



THE MOTION O.F A SHIP IN A SEAWAY 275 

(9.3.21) 

2 Q gP f xhdA-2 Qg p%A (y-y' c )MA-2 Q gp*yAxhdx-2QgPdzAx*hdx 

+o£ 2 [x(h y +d 11 )(s <p lx -(p u ) + +x{h y -6 11 )(s (p lx -(p lt )-]dA 

A 

We now assume that the propeller thrust T is of order ft 2 and is 
directed parallel to the <z'-axis: that is 

T = ft 2 T\' 

where i' is the unit vector along the #'-axis. We also assume that T is 
applied at a point in the longitudinal plane of symmetry of the ship / 
units below the center of mass. Thus we have the relations 

(9.3.22) T = ft 2 T\ + 0(/? 3 ), 
and 

(9.3.23) (R r - R c ) X T = - /j X T 

= lfi 2 Tk + O(pt). 

The mass of the ship is of order ft. If we write M = M x ft and ex- 
pand the left hand side of (9.2.18) in powers of ft it becomes 

i[MJ& % + M^'s, + 0(py\ + HM^y, + O(0»)] -k[0(/9«)] 

(9.3.24) = f pn dS + T - Mrfgl 

The expansion of the left hand side of (9.2.22) gives 

(9.3.25) l[0(jj»)] + j[0(/??)] + k[/ 3 #6 31 + O0»)] 

V(t - yd) X n dS + (R,. - R„) x T 



£ 



s 

where fil 31 is the moment of inertia of the ship about the axis which is 
perpendicular to the longitudinal plane of symmetry of the ship and 
which passes through the center of mass. 

If we replace the pressure integrals and thrust terms in the last two 
equations by (9.3.20), (9.3.21), (9.3.22), (9.3.23), and then equate the 
coefficients of like powers of ft in (9.3.24) and (9.3.25) we obtain the 
following linearized equations of motion of the ship. From the first 
order terms we find 



276 

(9.3.26) 



WATER WAVES 



(9.3.27) 2gg\ phdA=M 1 fig 

(9.3.28) xj3hdA=0 

(9.3.29) f [(s <p lx -(p lt ) + -(s o( p lx -(p lt )-]dA=0 

(9.3.30) [x(s q> lx -(p lt ) + -x(s (p lx -(p lt )-]dA = 

(9.3.31) | [(y-y' c )(s (p lx --(p lt ) + --(y-y' c ){s (p lx -(p lt )-]dA=O 
or by (9.3.29) 

(9.3.32) [yiSoV^-cpu^-yiSoy^-cpu^dA^O. 
From the second order terms we find 

(9.3.33) 

M 1 s 1 = q [{h x -d 21 )(s (p lx -(p lt ) + + (h x +d 21 )(s (p lx -(p lt )-]dA+T 

=Q [K{s (Pix-(Pit) + +h x {So(Pix-(Pit)~]d^+T, 

(9.3.34) 

M;y x =-2og[ (y^xd^hdx 

q [{K+®ii)( s oyix-nt) + +(K-Qi\)( s Mix-<Pit)-}dA 

2 Qg\ (yx+xO^hdx+g [h y (s (p lx -(p lt ) + +h y {s (p lx -(p lt )-]dA 



+, 



7 ai 91 =-20*0 



(y—y')MA 



-%Qgyi xhdx — 2ggd 31 x 2 hdx-\-lT 
-Q MK+^ii)( s o9ix—(Pit) + +x(K^iiK s o^ix—(Pit)~] dA 

Q l(y—y' c )(K—d2iKso<Pix—<Pity 

+(y-y' c )(i>.r+o 21 )(s (p lx -(p lt )-]dA 



THE MOTION OF A SHIP IN A SEAWAY 277 

or by (9.3.30), (9.3.31) 

hi0ai=-2Qg0*i f (y-y' c )hdA-2gg yi f xhdx-2 9 g6 31 f x*hdx+lT 
(9.3.35) 



+Q 



[xh y -{y-y')h x ][(s (p lx --(p lt ) + + {s (p lx -(p lt )-]dJ. 



Equation (9.3.26) states that the motion in the ^-direction is a 
small oscillation relative to a motion with uniform speed s = const. 
Equation (9.3.27) is an expression of Archimedes' law: the rest position 
of equilibrium must be such that the weight of the water displaced 
by the ship just equals the weight of the ship. The center of buoyancy 
of the ship is in the plane of symmetry, and equation (9.3.28) is an 
expression of the second law of equilibrium of a floating body; namely 
that the center of buoyancy for the equilibrium position is on the 
same vertical line, the y'-axis, as the center of gravity of the ship. 

The function q> x must satisfy 

<Plxx + <Plvv + <Plzz = ° 

in the domain D — A where D is the half space y<0, and A is the 
plane disk defined by the projection of the submerged hull on the 
x, y-plane when the ship is in the equilibrium position. We assume that 
A intersects the x, 2-plane. The boundary conditions at each side of A 
are 

(9 3 36) [ <P ^ = ~ S ^ hx ~ ° 2l) ~ (Wl + ® 21 ^ + ®^ y ~ y 'J 

\<piz= + s o( h x + 2 i ) - ( m i + 02i )# + 0n (2/ - y' c )- 

The boundary condition at y = is found by eliminating tj x from 
(9.3.10) and (9.3.11). Since w Q = these equations are 

- P]\ + s Mix —<Pit—$ 

- <Piv - V7i* + Vu = ° 
and they yield 

(9.3.37) s$<p lxx - 2s Q <p lxt + g(p ly + (p ltt = 

for y = 0. The boundary conditions (9.3.36) and (9.3.37) show that 
cp 1 depends on w^t), d lx (t) and 21 (t). The problem in potential theory 
for qp! can in principle be solved in the form 

<Pi = <Pi[x> V> z '> t'> w i(0» u (i), 21 {t)] 
without using (9.3.29), (9.3.30), (9.3.32). The significance of this has 



278 WATER WAVES 

already been discussed in sec. 9.1 in relation to equations (9.1.14). 
The general procedure to be followed in solving all problems was also 
discussed there. 

The remainder of this chapter is concerned with the special case of 
a ship which moves along a straight course into waves whose crests are 
at right angles to the course. In this case there are surging, heaving 
and pitching motions, but we have X = 0, d 2 = 0, co = 0; in addition 
we note that the potential function cp can be assumed to be an even 
function of z. Under these conditions the equations of motion are 
much simpler. They are 



r 1 s 1 =2 9 ( 



(9.3.38) M 1 s 1 = 2 e \ h x (s (p lx -(p lt )dA+T 



(9.3.39) M J y 1 =-2Qgy 1 \hdx-2Qgd 



1 



cchdcc J r 2Q h y (s (p lx —q) lt )dA 



A 

(9.3.40) / 81 03i=-2eg0 8 if {y-y c )hdA-2QgyA xhdx 
—2^31 I x 2 hdx+lT 



{ 



+2o [xh y — (y—y')h x ](s (p lx —(p lt )dA. 



It will be shown in the next section that an explicit integral represen- 
tation can be found for the corresponding potential function and that 
this leads to integral representations for the surge s v the heave y x 
and the pitching oscillation 6 31 . 



9.4. Method of solution of the problem of pitching and heaving of 
a ship in a seaway having normal incidence 

In this section we derive a method of solution of the problem of 
calculating the pitching, surging, and heaving motions in a seaway 
consisting of a train of waves with crests at right angles to the course 
of the ship, which is assumed to be a straight line (i.e., co = 0). The 
propeller thrust is assumed to be a constant vector. 

The harmonic function (p ± and the surface elevation r\ x therefore 
satisfy the following free surface conditions (cf. (9.3.10) and (9.3.11), 
with co = 0): 



THE MOTION OF A SHIP IN A SEAWAY 279 

(9.4.1) f-*h-+**.-** = = Q> 

\ - <Ply - Stflix +Vlt = ° 

The kinematic condition arising from the hull of the ship is (cf. 
(9.3.14) with 21 = n =0) 1 = 0): 

(9.4.2) (p lz = - s h x . 

Before writing down other conditions, including conditions at oo, 
we express cp ± as a sum of two harmonic functions, as follows 

(9.4.3) (p x (x, y, z; t) = Xo(x, y, z) + Xi( x > V> z > 0- 

Here % is a harmonic function independent of / which is also an even 
function of z. We now suppose that the motion of the ship is a steady 
simple harmonic motion in the time when observed from the moving 
coordinate system o — x,y, z. (Presumably such a state would result 
after a long time upon starting from rest under a constant propeller 
thrust.) Consequently we interpret Xo( x > V> z ) as the disturbance 
caused by the ship, which therefore dies out at oo; while Xii x > y, z; t) 
represents a train of simple harmonic plane waves covering the whole 
surface of the water. Thus X\ is given, with respect to the fixed coor- 
dinate system 0— X, Y, Z by the well-known formula (cf. Chapter 3): 



Xi 



Ce g sin lot + — X +y|, 



with g the frequency of the waves. In the o — x, y, z system we have, 
therefore : 

CT 2 

(9.4.4) Xl (x, y, z; t) = Ce^ sin |~- x + (a + *-^\ t + y 

We observe that the frequency, relative to the ship, is increased above 
the value a if s is positive — i.e. if the ship is heading into the waves 
—and this is, of course, to be expected. With this choice of Xv i* i s 
easy to verify that % satisfies the following conditions: 

(9.4.5) sfaoxx + gXoy = ° at y = 0, 
obtained after eliminating ?] 1 from (9.4.1), and 

(9.4.6) xoz = - SqK on A, 

with A, as above, the projection of the ship's hull (for z > 0) on its 
vertical mid-section. In addition, we require that % ->- at oo. 



280 WATER WAVES 

It should be remarked at this point that the classical problem con- 
cerning the waves created by the hull of a ship, first treated by Michell 
[M.9], Havelock [H.7], and many others, is exactly the problem of 
determining # from the conditions (9.4.5) and (9.4.6). Afterwards, 
the insertion of cp x = % Q in (9.3.38), with s x = 0, cp lt = 0, leads to the 
formula for the wave resistance of the ship — i.e. the propeller thrust T 
is determined. Since y 1 and 3 are independent of the time in this case, 
one sees that the other dynamical equations, (9.3.39) and (9.3.40), 
yield the displacement of the e.g. relative to the rest position of 
equilibrium (the heave), and the longitudinal tilt angle (the pitching 
angle). However, in the literature cited, the latter two quantities are 
taken to be zero, which implies that appropriate constraints would 
be needed to hold the ship in such a position relative to the water. 
The main quantity of interest, though, is the wave resistance, and it 
is not affected (in the first order theory, at least) by the heave and pitch. 

We proceed to the determination of # , using a method different 
from the classical method and following, rather, a course which it is 
hoped can be generalized in such a way as to yield solutions in more 
difficult cases. 

Suppose that we know the Green's function G*(£, rj, £; x,y,z) such that 
G* is a harmonic function for r\ < 0, £ > except at (x, y, z) where 
it has the singularity 1/r; and G* satisfies the boundary conditions 
(9.4.7) G* + kG* = on r) = 

G* = onC = 
where k = g/s*. We shall obtain this function explicitly in a moment, 
and will proceed here to indicate how it is used. Let Z denote the half 
plane r\ = 0, £ > 0; and let Q denote the half plane £ = 0, yj < 0. 
From Green's formula and the classical argument involving the 
singularity 1/r we have 

***»= " (j X oG*mt+jjxonG*dgdZ-jjxofi*d£dt). 



Then, since 



N X «G*d$di; + [Lofi*d£dZ = iJJ (X*G%- Xu fi* )<IWH 



1 

0, 



THE MOTION OF A SHIP IN A SEAWAY 281 

we have an explicit representation of the solution in the form 
Xo(v> V> z ) = — — Xofi*didr), or 

Q 

(9.4.8) Xq (x, y, z) = fl [[h(£, Tj)G*(£ 9 rj, 0; x, y, z) d£ drj, 

A 

upon using (9.4.6). 

In order to determine G* consider the Green's function G(£, rj, £; 
x, y, z) for the half space rj < which satisfies 

G« + kG v = 

on rj = 0. This function can be written as 

where 



r i V(£-x)*+(v-y) 2 +(t-z) 2 
i i 

r 2 V(^-^) 2 + (77+2/) 2 + (C-^) 2 



and g is a potential function in rj < which satisfies 

a i 



dy V(e - xf + i/ 2 + (t - 2) 2 

on 37 = 0. The formula 

(obtained from the well-known analogous representation for 1/r) in 
which the Bessel function J can be expressed as 

2 r n/2 

J r ? ?V(^-a:) 2 + (C-2) 2 ]= — cos [p(£-x) cos 0] cos [p(f-«) sin 6] dd, 

allows us to write 

g&+hg n = — p^^ cos [p(£— #) cos 0] cos [p(£—z) sin 0] d0 dp 

n J J 

for 7/ = and y < 0. It is now easy to see that 



282 WATER WAVES 

4& f °° C nl2 
gg+kg n = — V e^v+n) cos [p(£— a?) cos 0] cos [>(£—*) sin 0] dfl dp 

^ J J 
is a potential function in r\ < which satisfies the boundary condition. 
An interchange of the order of integration gives 

4k M 2 f °° 

gii J r kg n = — dJd 0te\ p cos Ip(t-z) sin 0]«»[(»+*) +*(*-«) cos ] dp 

^ J Jo 

where ^ denotes the real part. If we think of p as a complex variable, 
the path from to 00 in the last result can be replaced by any equi- 
valent path L, to be chosen later: 

4<k M 2 C 

g H +kg n = _ dd&e\ p cos [p(C-z) sin 0]e*t (»+»>)+<(*-*) «* 0] dp> 

^Jo Jl 

Since the right hand side of this differential equation for g is expressed 
as a superposition of exponentials in £ and rj it is to be expected that 
a solution of it can be found in the form 

g= *1 r' 2 d6 me [ pC ° s[ ^-* )sin0] ^<*W+«M -so] dp 
ttJ J l kp—p 2 cos 2 6 

provided the path L can be properly chosen. The path L, which will 
be fixed by a condition given below, must, of course, avoid the pole 
p = k/cos 2 6. 

It can now be seen that the function 6?*(£, rj, £; x, y, z) = 
G(£, rj, f J x, y> z) + G(£, rj, J; x, y, — z) satisfies all the conditions 
imposed on the Green's function employed in (9.4.8): the sum on the 
right has the proper singularity in rj < 0, £ > 0, it satisfies the 
boundary condition (9.4.7) and 

G c (g, rj, C; x, y, z) + G^, rj, f ; a?, t/, — 2) 
is zero at £ = 0. Thus we have for G* the representation: 

1 1 1 



G* 



f=o 



8A; r /2 JD si* r c ° s (p* sin 0) ^^h*^-*) cose ] dp 

— au we — 

n J Jl k—p cos 2 6 



The substitution of this in (9.4.8) gives finally 

Xo(^y^)=~ fffrtf. i?)f-7= * — 1 =W*? 

2 ^JJ (-v/(f-a;) a + ()y-y) s +z a \ / (£-a:) 1! +(»?+y) s +2 t J 

+ ?4 ff^, ,) ( f W f cos( P ;sin 9 )^+^-*>-^ j ^ 
7T 2 JJ U Jl k-p cos 2 6 J 



THE MOTION OF A SHIP IN A SEAWAY 



283 



A condition imposed on % Q (x, y, z) is that % Q (x, y, z) -> as x -> + c0 - 
This condition is satisfied if we take L to be the path shown in Fig. 9.4.1. 



(P) 



c/cos 2 



Fig. 9.4.1. The path L in the p-plane, with c = k 
The function cp ± is given by 



<Pi = Xi + Xo = Ce 9 sin 



ig \ g t 



+ y 



+ Xo 



and therefore the important quantity s (p lx — cp lt is given by 



\t+y 



i s oZo*' 



(9.4.9) s (p lx - <p lt = - Cw * cos — + (ct -f — ) 

L g \ g t 

If this is substituted in the equation (9.3.38) for the surge we have 



M 1 s 1 =—2qCg \\h x e 9 cos 



a \\h x e * cos \— + U+— \t+y dxdy 



+2Q*o WKZoxdvdy+T. 



*J> 



The last equation shows that in order to keep s x bounded for all t we 
must take for T the value 



(9.4.10) T = -2gs ((h xZox dx 



dy 



where 



Xox 



tey.o)=±[[w,f,)l r „ tf" g) 



2ttJJ 

25 



«-*: 



[(|-^) 2 + (^-2/) 2 ] 3/2 [(£-*) 2 + (>?+*/) 2 ] 3/2 

Igp COS e v[(v+V)+i(^-x) cos 6] ^ | 



^h^ 9 r])lr l2 dd^e( 



g~ s oP cos2 



dt-drj 
dg drj. 



284 WATER WAVES 

In effect, T is determined by the other time-independent term in the 
equation of motion. Equation (9.4.10) gives the thrust necessary to 
maintain the speed s , or inversely it gives the speed s which corres- 
ponds to a given thrust. The integral in (9.4.10) is called the wave 
resistance integral. As one sees, it does not depend on the seaway. The 
integral can be expressed in a simpler form as follows. 
The function %$ x {x, y, 0) is a sum of integrals of the type 



I! 



h $ (g,r))f(§ 9 ri; x, y) dgdrj. 



A 

If an integral of this type is substituted in the wave resistance integral 
we have 

h x (x, y)ht(£, r])f(g, r\\ x, y) d£drjdxdy=I 

4 A 

say. This is the same as 

h $ (g, rj)h x (x, y)f(x, y;g,rj) dxdyd£dr]=I 

A A 

and we see that / = if 



f(i,r];x, y) = — f(x, y; g t rj). 
Therefore 

T= ^ JT jjh x (x, y)h^, n )h d^dxdy 

A A 

where 

■«/2 r ig p cos d e p{v+r)) cos [p(g—x) cos 6] dp 



/ 1= = dd&e 



g—sftp cos 2 6 

Since &e is zero except for the residue from the integration along 
the semi-circular path centered at the point 



s% cos 2 6 cos 2 6 
we find from the evaluation of this residue that 

f 1= ^L P sec 3 6 e k{y + ^ sec20 cos [h(g-x) cos 6] dd. 
4 U 



285 



THE MOTION OF A SHIP IN A SEAWAY 

We introduce Michell's notation: 

P(d)= [[h x {x, y)e kysec2d cos (kx sec 6) dxdy 

A 

Q(0) = Uh x (x 9 y)e kv sec20 sin (kx sec 0) dxdy 

A 

and can then write 

T= -**- (P 2 +Q 2 ) sec 3 6 dd. 
^o Jo 

This is the familiar formula of Michell for the wave resistance. 
The surge is given by 

a 2 y r- 9 



fe 



~ 2 ^ ff^ sin K 8+g+ ^!U + 

<r+ V *)AfJJ Lg \ g / 



dxdy. 



Hereafter we will suppose for simplicity that there is no coupling 

between (9.3.39) and (9.3.40), so that xhdx = 0. The substitution 

of (9.4.9) in (9.3.39) therefore gives the following equation for the 
heave : 



Mm + 



2gg i hdx y Y =— 2qCg\ 



h y e 9 cos 



: g \ g J 

o K%ox dxdy. 



t+y 



dxdy 



The time independent part of y v the heave component of the trim, 
we denote by yf ; it is given by 

(9.4.11) Igl hdx\y*=s \h yXox dxdy. 

A 

Here y* is the vertical displacement of the center of gravity of the 
ship from its rest position when moving in calm water. The integral 
on the right hand side of (9.4.11) is even more difficult to evaluate 
than the wave resistance integral. 

The response to the seaway in the heave component is given by 



286 



WATER WAVES 



yf* 



2qCg \\h y e 9 cos — + /or+^ — J £ + 7 

A 

2 Q g[ hdx-~M 1 (a+ S ^ 2 \ 



dxdy 






For the case under consideration, the theory predicts that resonance 
in the heave occurs when 



s n a d 



CT+ — = 



'2Qg 



hdx 



The equation for the pitching angle is 



V 2 



hAi+ZQg 



(y—y' e ) hdA + x * hdx 

= ~2qCo \\[cch y -(y-y' e )h x ] cos J — + /<j + — ) * + y ] dxdy 

A 

+IT+2qs \ [xh y ~(y~y' c )h x }x 0x dA. 
The time independent part of 6 31 , which we denote by 6 31 is given by 



2?g 



ft* 



f (y-y' c )hdA + J x 2 hdx 
=IT+2qs \ [xh y -(y-y' c )h x ]x 0x dA 

= (l-y' c )T+2QsA [xh y -yh x ]x 0x dA. 



The angle ^ is called the angle of trim; it is the angular displacement 
of a ship which moves with the speed s in calm water. 
The oscillatory part of the heave d zl to the sea is 

-2qCg [xh y — (y—y' c )h x ] cos — + lo+ — \t + y dxdy 



'31 



*Qg 



f (y-y' c )hdA+[x*hdx -lJ a + S -^\ 



and we see that the theory predicts resonance when 



<? + 



S n <7< 



2Qg 

I 



31 



:i 



{y-y' e ) hdAJ r 



x 2 hdx 



V. 



THE MOTION OF A SHIP IN A SEAWAY 287 

Of course, the differential equations for y x and d 31 permit also 
solutions of the type of free undamped oscillations of a definite fre- 
quency (in fact, having the resonant frequencies just discussed) but 
with arbitrary amplitudes which could be fixed by appropriate initial 
conditions. This point has been discussed at length in the introduction 
to this chapter. 



PART III 



CHAPTER 10 



Long Waves in Shallow Water 



10.1. Introductory Remarks and Recapitulation of the Basic Equations 

The basic theory for waves in shallow water has already been de- 
rived at length in Chapter 2 in two different ways: one derivation, 
along conventional lines, proceeded on the basis of assuming the 
pressure to be determined by the hydrostatic pressure law p = 
gg(r) — y) (see Fig. 10.1.1), the other by making a formal develop- 
ment in powers of a parameter g; the two theories are the same in 



Free Surface 




Bottom 



Fig. 10.1.1. Long waves in shallow water 



lowest order. With one exception, the present chapter will make use 
only of the theory to lowest order and consequently the derivation of 
it given in sections 2 and 3 of Chapter 2 suffices for all sections ol this 
chapter except section 9. 

We recapitulate the basic equations. In terms of the horizontal 
velocity component u = u(x, t), and the free surface elevation 
T) =rj(x,t) the differential equations (cf. (2.2.11), (2.2.12)) are 

(10.1.1) u t + uu x = — grj x , 

(10.1.2) [u(rj + h)] x = - Tj t . 

291 



292 WATER WAVES 

It is sometimes useful and interesting to make reference to the gas 
dynamics analogy, by introducing the "density" q through 

(10.1.3) q = g(rj +h) 9 

and the "pressure" p by p = pdy, which in view of the hydrostatic 

pressure law yields the relation 

(10.1.4) p = i ?- 

This is an "adiabatic law" with "adiabatic exponent" 2 connecting 
pressure and density. As one sees, it is the depth of the water, essen- 
tially, which plays the role of the density in a gas. In terms of these 
quantities, the equations (10.1.1) and (10.1.2) take the form 

(10.1.5) g(u t + uu x ) = - p x + ggh x , 

(10.1.6) (gu) x = - Q t . 

These equations, together with (10.1.4), correspond exactly to the 
equations of compressible gas dynamics for a one-dimensional flow if 
h x = 0, i.e. if the depth of the undisturbed stream is constant. It 
follows that a "sound speed" or propagation speed c for the pheno- 
mena governed by these equations is defined by c = Vdp/dg, as in 
acoustics, and this quantity in our case has the value 

(10.1.7) c = Y~ = Vg( V +h), 

as we see from (10.1.4) and (10.1.3). Later on, we shall see that it is 
indeed justified to call the quantity c the propagation speed since it 
represents the local speed of propagation of "small disturbances" 
relative to the moving stream. We observe the important fact that c 
(which obviously is a function of a: and /) is proportional to the square 
root of the depth of the water. 

The propagation speed c(x, t) is a quantity of such importance that 
it is worthwhile to reformulate the basic equations (10.1.1 ) and (10.1.2) 
with c in place of rj. Since c x = (grj x + ^h x )j2c and c t = grj t l2c one 
finds readily 

(10.1.8) u t + uu x + 2cc x — H x = 0, 

(10.1.9) 2c, + 2uc x + cu x = 0, 
with 

(10.1.10) H = gh. 



LONG WAVES IN SHALLOW WATER 293 

The verification in the general case that the quantity c represents 
a wave propagation speed requires a rather thorough study of certain 
basic properties of the differential equations. However, if we restrict 
ourselves to motions which depart only slightly from the rest position 
of equilibrium (i.e. the state with r\ = 0, u = 0) it is easy to verify 
that the quantity c then is indeed the propagation speed. From (10.1.7) 
we would have in this case c = c + e(x, t), with c = vgh and e a 
small quantity of first order. We assume u and its derivatives also to 
be small of first order and, in addition, take the case in which the 
depth h is constant. Under these circumstances the equations (10.1.8) 
and (10.1.9) yield 

(10.1.11) u t + 2c e x = 0, 

(10.1.12) 2e t + c u x = 

if first order terms only are retained. By eliminating s we obtain for 
u the differential equation 

(10.1.13) u tt - c%u xx = 0. 

This is the classical linear wave equation all solutions of which are 
functions of the form u = u{x i c t) and this means that the motions 
are superpositions of waves with constant propagation speed c = Vgh. 
The role of the quantity c as a propagation speed (together with 
many other pertinent facts) can be understood most readily by dis- 
cussing the underlying integration theory of equations (10.1.8) and 
(10.1.9) by using what is called the method of characteristics; we 
turn therefore to a discussion of this method in the next section. 



10.2. Integration of the Differential Equations by the Method of 
Characteristics 

The theory of our basic differential equations (10.1.8) and (10.1.9), 
which are of the same form as those in compressible gas dynamics, 
has been very extensively developed because of the practical necessity 
for dealing with the flow of compressible gases. The purpose of the 
present section is to summarize those features of this theory which 
can be made useful for discussing the propagation of surface waves 
in shallow water. In doing so, extensive use has been made of the 
presentation given in the book by Courant and Friedrichs [C.9]; in 
fact, a good deal of the material in sections 10.2 to 10.7, inclusive, 
follows the presentation given there. 



294 WATER WAVES 

The essential point is that the partial differential equations (10.1.8) 
and (10.1.9) are of such a form that the initial value problems asso- 
ciated with them admit of a rather simple discussion in terms of a 
pair of ordinary differential equations called the characteristic differ- 
ential equations. We proceed to derive the characteristic equations 
for the special case in which [cf. (10.1.10)] 

(10.2.1) H x = m = const. 

i.e. the case in which the bottom slope is constant. In fact, this is the 
only case we consider in this chapter. If we add equations (10.1.8) 
and (10.1.9) it is readily seen that the result can be written in the 
form: 

id d \ 

(10.2.2) ! - + (u + c) j^ J . (u + 2c - mt) = 0. 

The expression in brackets is, of course, to be understood as a dif- 
ferential operator. Similarly, a subtraction of (10.1.9) from (10.1.8) 
yields 

d a 



(10.2.3) \—. + (u - c) j- • (u - 2c - mt) = 0. 

But the interpretation of the operations defined in (10.2.2) and 
(10.2.3) is well known (cf. (1.1.3)): the relation (10.2.2), for example, 
states that the function (u + 2c — mt) is constant for a point moving 
through the fluid with the velocity (u -f- c), or, as we may also put it, 
for a point whose motion is characterized by the ordinary differential 
equation dx/dt --- u + c. Equation (10.2.3) can be similarly interpre- 
ted. That is, we have the following situation in the cc, /-plane: There are 
two sets of curves, C x and C 2 , called characteristics, which are the 
solution curves of the ordinary differential equations 



(10.2.4 



dx 
dt 
dx 



C, : — - = u + c, and 
dt 



C 2 : — — u — c, 



dt 
and we have the relations 

u + 2c — mt = k x = const, along a curve C x and 



(10.2.5) 

u — 2c — mt — k 2 = const, along a curve C 2 . 

Of course the constants k 1 and k 2 will be different on different curves 
in general. It should also be observed that the two families of charac- 



LONG WAVES IN SHALLOW WATER 295 

teristics determined by (10.2.4) are really distinct because of the fact 
that c = Vg(r) + h) ^ since we suppose that r\ > — h, i.e. that 
the water surface never touches the bottom. 

By reversing the above procedure it can be seen rather easily that 
the system of relations (10.2.4) and (10.2.5) is completely equivalent 
to the system of equations (10.1.8) and (10.1.9) for the case of con- 
stant bottom slope, so that a solution of either system yields a solu- 
tion of the other. In fact, if we set f(x, t) = u + 2c — mt and ob- 
serve that f(x, t) = k ± = const, along any curve x = x(t) for which 
dx/dt = u -+- c it follows that along such curves 

dx 

(10.2.6) ft J rf x -^ = ft + (u+ c)f x = 0. 

In the same way the function g(x, t) = u — 2c — mt satisfies the 
relation 

(10.2.7) g t + ■ (u - c)g x = 

along the curves for which dx/dt = u — c. Thus wherever the curve 
families C 1 and C 2 cover the x, f-plane in such a way as to form a non- 
singular curvilinear coordinate system the relations (10.2.6) and 

(10.2.7) hold. If now equations (10.2.6) and (10.2.7) are added and 
the definitions of f(x, t) and g(x, t) are recalled it is readily seen that 
equation (10.1.8) results. By subtracting (10.2.7) from (10.2.6) 
equation (10.1.9) is obtained. In other words, any functions u and c 
which satisfy the relations (10.2.4) and (10.2.5) will also satisfy 

(10.1.8) and (10.1.9) and the two systems of equations are therefore 
now seen to be completely equivalent. 

As we would expect on physical grounds, a solution of the original 
dynamical equations (10.1.8) and (10.1.9) could be shown to be 
uniquely determined when appropriate initial conditions (for t = 0, 
say) are prescribed; it follows that a solution of (10.2.4) and (10.2. 
5 ) is also uniquely determined when initial conditions are prescribed 
since we know that the two systems of equations are equivalent. 

At first sight one might be inclined to regard the relations (10.2.4) 
and (10.2.5) as more complicated to deal with than the original dif- 
ferential equations, particularly since the right hand sides of (10.2.4) 
are not known in advance and hence the characteristic curves are also 
not known: they must, in fact, be determined in the course of deter- 
mining the unknown functions u and c which constitute the desired 
solution. Nevertheless, the formulation of our problems in terms of 



296 



WATER WAVES 



the characteristic form is quite useful in studying properties of the 
solutions and also in studying questions referring to the appropriate- 
ness of various boundary and initial conditions. It is useful to begin 
by describing briefly a method of determining the characteristics and 
thus the solution of a given problem by a method of successive approx- 
imation which at the same time makes possible a number of useful 
observations and interpretations regarding the role played by the 
characteristics in general. Let us for this purpose consider a problem 
in which the values of the velocity u and the surface elevation r\ 
(or, what amounts to the same thing, the propagation or wave speed 

c = Vg(r] + h)) are prescribed for all values of x at the initial instant 
t = 0. We wish to calculate the solution for t > by determining u 
and c through use of (10.2.4) and (10.2.5) and the given initial condi- 
tions. At t = we assume that 



(10.2.8) 



u(x, 0) = u(x) 
c(x, 0) = ~c(x) 

in which u(x) and c(x) are given functions. We can approximate the 
values of u and c for small values of t as follows : consider a series of 
points on the #-axis (cf. Fig. 10.2.1) a small distance dx apart.. At all 
of these points the values of u and c are known from (10.2.8). Conse- 
quently the slopes of the characteristics C x and C 2 at these points are 




8x ' Sx i Sx 
Fig. 10.2.1. Integration by finite differences 

known from (10.2.4). From the points 1, 2, 3, 4 straight line segments 
with these slopes are drawn until they intersect at points 5, 6, and 7, 
and if dx is chosen sufficiently small it is reasonable to expect that 



LONG WAVES IN SHALLOW WATER 297 

the positions of these points will be good approximations to the inter- 
sections of the characteristics issuing from the points 1, 2, 3, 4 since 
we are simply replacing short segments of these curves by their 
tangents. The values of both x and t at points 5, 6, and 7 are now 
known— they can be determined graphically for example— and 
through the use of (10.2.5) and the initial conditions we can also 
determine the approximate values of u and c at these points. For this 
purpose we observe that along any particular segment issuing from 
the points 1, 2, 3 or 4 the values of u + 2c — mt and u — 2c — mt 
are known constants since the values of u and c are fixed by (10.2.8) 
for t = 0; hence we have 

( along C\ : u + 2c — mt = u -f 2c, and 
(10.2.9) \ 1 J - - 

{ along C 2 : u — 2c — mt = u — 2c. 

At the points 5, 6, and 7 we know the values of t and hence (10.2.9) 
furnishes two independent linear equations for the determination of 
the values of u and c at each of these points. Once u and c are known 
at points 5, 6, and 7 the slopes of the characteristics issuing from these 
points can be determined once more from (10.2.4) and the entire pro- 
cess can be carried out again to yield the additional points 8 and 9 
and the approximate values of u and c at these points. In this way 
we can approximate the values of u and c at the points of a net over 
a certain region of the x, Z-plane, and can then obtain approximate 
values for u and c at any points in the same region either by inter- 
polation or by refining the net inside the region. It is quite plausible 
and could be proved mathematically that the above process would 
converge as bx -> to the unique solution of (10.2.4) and (10.2.5) 
corresponding to the given initial conditions for sufficiently small 
values of t (i.e. for a region of the x, Z-plane not too far from the cr-axis) 
provided that the prescribed initial values of u and c are sufficiently 
regular functions of x— for example, if they have piecewise con- 
tinuous first derivatives. 

It should be clear that once the characteristics are known the values 
of u and c for all points of the x, £-plane covered by them are also 
known, since the constants k x and k 2 in (10.2.5) are known on each 
characteristic through the initial data and hence the values of u and c 
for any point (x, t) can be calculated by solving the linear equation 
(10.2.5) for the characteristics through that point. This statement of 
course implies that each one of the two families of characteristics 
covers a region of the x, 2-plane simply and that no two members of 



298 



WATER WAVES 



different families are tangent to each other — in other words it is 
implied that the two families of characteristics form a regular curvi- 
linear coordinate system over the region of the x, Z-plane in question. 
One of the points of major interest in the later discussion centers 
around the question of determining where and when the character- 
istics cease to have this property, and of interpreting the physical 
meaning of such occurrences. 

The method of finite differences used above to determine the cha- 
racteristics can be interpreted in such a way as to throw a strong light 
on the physical properties of the solution. Consider the point 10 of 
Fig. 10.2.1 for example. We recall that the approximate values u 10 
and c 10 of u and c at point 10 were obtained through making use of the 
initial values of u and c at points 1, 2, 3, 4 on the ^-axis only, and 
furthermore that the values u 10 and c 10 required the use of points con- 
fined solely to the region within the approximate characteristics join- 
ing point 10 with points 1 and 4. Since the finite difference scheme 
outlined above converges as dx -> to yield the exact characteristics 
we are led to make the following important statement: the values of u 
and c at any point P(x, t) within the region of existence of the solution 
are determined solely by the initial values prescribed on the segment of 
the x-axis which is subtended by the two characteristics issuing from P. 





Domain of 
determinacy 



Range of influence of Q Domain of dependence of P 

Fig. 10.2.2. Domain of dependence and range of influence 



In addition, the two characteristics issuing from P are also determined 
solely by the initial values on the segment subtended by them. Such 
a segment of the a>axis is often called the domain of dependence of the 



LONG WAVES IN SHALLOW WATER 



299 



point P. Correspondingly we may define the range of influence of a 
point Q on the #-axis as the region of the x, /-plane in which the values 
of u and c are influenced by the initial values assigned to point Q. In 
Fig. 10.2.2 we indicate these two regions. It is also useful on occasion 
to introduce the notion of domain of determinacy relative to a given 
domain of dependence. It is the region in which the motion is deter- 
mined solely by the data over a certain segment of the .r-axis. These 
regions are outlined by characteristic curves, as indicated in Fig. 
10.2.2, in an easily understandable fashion in view of the discussion 
above. 

We are now in a position to understand why it is appropriate to 
call the quantity c the propagation or wave speed. To this end we 
suppose that a certain motion of water exists at a definite time, which 
we take to be t = 0. This means, of course, that u and c are known at 
that time, and, as we have just seen, the motion would be uniquely 
determined for t > 0. However, we raise the question: what difference 
would there be in the subsequent motion if we created a disturbance 
in some part of the fluid, say over a segment Q ± Q 2 of the #-axis (cf. 
Fig. 10.2.3)? This amounts to asking for a comparison of two solutions 
of our equations which differ only because of a difference in the initial 




Fig. 10.2.3. Propagation of disturbances 



conditions over the segment Q X Q 2 . Our whole discussion shows that the 
two solutions in question would differ only in the shaded region of 
Fig. 10.2.3, which comprises all points of the x, /-plane influenced by 
the data on the segment Q ± Q 2 > an d which is bounded by characteristics 
C 2 and C x issuing from the endpoints Q x and Q 2 of the segment. These 



curves, 
dx'dt = 



however, satisfy the differential equations dx/dt = u — c, 
u -f c. Since u represents the velocity of the water, it is then 



300 WATER WAVES 

clear that c represents the speed relative to the flowing stream at 
which the disturbance on the segment Q X Q 2 spreads over the water. 
This implies that the data in our two problems really differ at points 
Qj and Q 2 and that these differences persist along the characteristics 
issuing from these points. Actually, only discontinuities in derivatives 
at Q ± and Q 2 (and not of the functions themselves) are permitted in the 
above theory, and it could be shown that such discontinuities would 
never smooth out entirely along the characteristics C 1 and C 2 . We are 
therefore justified in referring to the quantity c = Vg(n + h) as the 
(local) propagation speed of small disturbances— that is, small in the 
sense that only discontinuities in derivatives occur at the front of a 
disturbance. 

10.3. The Notion of a Simple Wave 

There is an important class of problems in which the theory of 
characteristics as presented in the preceding section becomes parti- 
cularly simple. These are the problems in which (1) the initial un- 
disturbed depth h of the water is constant so that the quantity m in 
(10.2.1) (cf. also (10.1.10)) is zero, (2) the water extends from the 
origin to infinity at least in one direction, say in the positive direction 
of the #-axis, and (3) the water is either at rest or moves with constant 
velocity and the elevation of its free surface is zero at the time t = 0. 
In other words, the water is in a uniform state at time t = such that 
u = u = const, and c = c = Vgh — const, at that instant. Our 
discussion from here on is modeled closely on the discussion given by 
Courant and Friedrichs [C.9], Chapter III. 

We now suppose that a disturbance is initiated at the origin x = 
so that either the particle velocity u, or the surface elevation rj (or the 
wave velocity c = Vg(rj -f- h)) changes with the time in a prescribed 
manner.* That is, a disturbance at one point in the water propagates 
into water of constant depth and uniform velocity. Under these 
circumstances we show that one of the two families of characteristics 
furnished by (10.2.4) consists entirely of straight lines along each of which 
u and c are constant. The corresponding motion we call a simple wave. 

* One might accomplish this experimentally in a tank as follows: To obtain 
a prescribed velocity u at one point it would only be necesary to place a vertical 
plate in the water extending from the surface of the water to the bottom of the 
tank and to move it with the prescribed velocity. To change rj at one point water 
might be either poured into the tank or pumped out of it at that point at an 
appropriate rate. 



LONG WAVES IN SHALLOW WATER 



301 



Our statement is an immediate consequence of the following funda- 
mental fact: if the values of u and c on any characteristic curve, Cj say 
(i.e. a solution curve of the first of the two ordinary differential equations 
(10.2.4)), are constant, then Cj is a straight line and furthermore it is 
embedded in a family of straight line characteristics along each of which 
u and c are constant, at least in a region of the x, /-plane where u(x, t) 
and c(x, t) are without singularities and which is covered by the 
two distinct families of characteristics. The proof is easily given. To 
begin with, the curve Cj is a straight line if u and c are constant along 
it, since the slope of the curve is constant in that case from (10.2.4). 
Next, let C x be another characteristic near to Cj. We consider any two 
points A and B on Cj together with the characteristics of the family 
C 2 through A and B and suppose that the latter characteristics 
intersect C x at points A and B (cf. Fig. 10.3.1 ): To prove our statement 




Fig. 10.3.1. Region containing a straight characteristic 



we need only show that u (A) = u(B) and c(A) = c(B) since then u and c 
would be constant on C ± (because of the fact that A and B are any 
arbitrary points on C x ) and hence the slope of the curve C x would be 
constant, just as was argued for CJ. We have u(A ) = u(B ) and 
c(A ) = c(B ) and consequently we may write 



(10.3.1) 



u A - 2c A 
u B — 2c B 



U>A, 



2e d 

2c j. 



2c 



by making use of the second relation of (10.2.5) (which holds along 
the characteristics C 2 ) and observing that m = since the original 



302 



WATER WAVES 



depth of the water is assumed to be constant. Next we make use of 
the first relation of (10.2.5) for C ± to obtain 

(10.3.2) u A + 2c A = u B + 2c B . 
But from (10.3.1) we have 

(10.3.3) u A - 2c A =u B - 2c B , 

and (10.3.2) and (10.3.3) are obviously satisfied only if u A = u B 
and c A = c B . Our statement is therefore proved. 

The problems formulated in the first paragraph of this section are 
at once seen to have solutions (at least in certain regions of the 
x, 2-plane) of the type we have just defined as simple waves since 
there is a region near the a?-axis in the x, /-plane throughout which the 
particle velocity u and wave speed c are constant, and in which there- 
fore the characteristics are two sets of parallel straight lines. The cir- 
cumstances are illustrated in Fig. 10.3.2 below: There is a zone/ along 



t=T 




Fig. 10.3.2. A simple wave 



the a?-axis which might be called the zone of quiet* inside which the 
characteristics are obviously straight lines x i c t = const. (These 
lines are not drawn in the figure). This region is terminated on the 
upper side by an "initial characteristic" x = c t which divides the 

* In a "zone of quiet" we permit the particle velocity u to be a non zero 
constant, but the free surface elevation r/ is taken to be zero in such a region. 
In case u = u = const. ^ initially, the motion can be thought of as observed 
from a coordinate system moving with that velocity; thus there is no real loss 
of generality in assuming u = 0, as we frequently do in the following. 



LONG WAVES IN SHALLOW WATER 303 

region of quiet from the disturbed region above it. The physical inter- 
pretation of this is of course that the disturbance initiated at the 
time t — propagates into the region of quiet, and the water at any 
point remains unaffected until sufficient time has elapsed to allow 
the disturbance to reach that point. The exact nature of the motion 
in the disturbed region is determined, of course, by the character of 
the disturbance created at the point x = 0, i.e., by appropriate data 
prescribed along the /-axis.* One set of characteristics, i.e., the set 
containing the initial characteristic C?, therefore consists of straight 
lines. (That the characteristics C 2 in the zone// are necessarily curved 
lines and not straight lines can be seen from the fact that they would 
otherwise be the continuations of the straight characteristics from the 
zone / of quiet and hence the zone // would also be a zone of quiet, as 
one sees immediately). Furthermore, the set of straight characteristics 
C x in zone // is completely determined by appropriate conditions pre- 
scribed at x = for all /, i.e., along the /-axis. What these conditions 
should be can be inferred from the following discussion. Consider any 
straight characteristic issuing from a point / = r on the /-axis. We 
know that the slope dx/dt of this straight line is given in view of 
(10.2.4), by 

(10.3.4) ~J t =U[x) +C(t) ' 

Suppose now that there is a curved characteristic C 2 going back 
from / = r on the /-axis to the initial characteristic Cj (see the dotted 
curve in Fig. 10.3.2). We have the following relation from (10.2.5): 

(10.3.5) u(r) — 2c(t) = u — 2c , 

in which u and c are the known values of u and c in the zone of quiet. 
Hence the slope of any of the straight characteristics issuing from the 
/-axis can be given in either of the two forms: 

dx 1 



(10.3.6) 



— == — [3u(r) - u ] + c , or 

dx 

3c(t) — 2c + u , 



{ dt 

as one sees from (10.3.4) and (10.3.5). Thus if either u(r) or c(r) is 

* Our discussion in the preceding section centered about the initial value 
problem for the case in which the initial data are prescribed on the ai-axis, but one 
sees readily that the same discussion would apply with only slight modifications 
to the present case, in which what is commonly called a boundary condition (i.e. 
at the boundary point x = 0), rather than an initial condition, is prescribed. 



304 



WATER WAVES 



given, i.e. if either u or c is prescribed along the Z-axis, then the slopes 
of the straight characteristics C 1 and with them the characteristics C x 
themselves are determined. Since we know, from (10.3.5), the values 
of both u and c along the £-axis if either one is given, and since u and c 
are clearly constant along the straight characteristics, it follows that we 
know the values of u and c throughout the entire disturbed region — in 
other words, the motion is completely determined. 

So far, we have considered only the case in which the curved cha- 
racteristics (i.e., those of the type C 2 ) which issue from the boundary 
x = c t of the disturbed region actually reach the 2-axis. This, however, 
need not be the case. Suppose, for example, that u is positive and 
u > c = Vgh. In this case the slope dx/dt of the curves C 2 is positive, 
and we cannot expect that they will turn to the left, as in Fig. 10.3.2. 
Indeed, in such a case one does not expect that a disturbance will pro- 
pagate upstream (that is, to the left in our case) since the stream velo- 
city is greater than the propagation speed. In gas dynamics one would 
say that the flow is supersonic, while in hydraulics the flow is said 
to be supercritical. One could also look at the matter in another way: 
For not too large values of t the velocity u can be expected to remain 
supersonic and hence for such values of t both sets of characteristics 
issuing from the t-axis would go into the right half plane (u being 
again taken positive). Thus we would have the situation indicated 
in Fig. 10.3.3, in which a segment of the Z-axis is subtended by two 





Fig. 10.3.3. The supercritical case 



LONG WAVES IN SHALLOW WATER 305 

characteristics drawn backward from P. In this case, as in the case of 
the initial value problem treated in the preceding section, we must 
prescribe the values of both u and c along the /-axis. If we do so, then 
the solution is once more determined through (10.3.4) and the fact 
that u + 2c is constant along one set of characteristics and u — 2c is 
constant along the other. 

In either of our two cases, i.e. of subcritical or supercritical flow, 
we see therefore that the simple wave can be determined. One sees 
also how useful the formulation in terms of the characteristics can be 
in determining appropriate subsidiary conditions such as boundary 
conditions. 

If we wish to know the values of u and c for any particular time 
t = t , once the simple wave configuration is determined, we need 
only draw the line t = t and observe its intersections with the 
straight characteristics since the values of u and c are presumably 
known on each one of the latter. Thus u and c would be known as 
functions of x for that particular time. Of course, the surface elevation 
rj would also be known from 

c = Vg(h +rj). 

10.4. Propagation of disturbances into still water of constant depth 

In the preceding section we have seen how the method of charac- 
teristics leads to the notion of a simple wave in terms of which we can 
describe with surprising ease the propagation of a disturbance initiat- 
ed at a point into water of constant depth moving with uniform speed. 
In the present section we consider in more detail the character of the 
simple waves which occur in two important special cases. We assume 
always that the pulse is initiated at x = and that it then propagates 
in the positive ^-direction into still water. Thus we are considering 
cases in which the flow is subcritical at the outset. 

One of the most striking and important features of our whole dis- 
cussion is that there is an essential difference between the propagation 
of a pulse which is created by steadily decreasing the surface elevation 
yj at x = and of a pulse which results by steadily increasing the ele- 
vation at x = 0. If the pulse is created by initiating a change in the 
particle velocity u at x = (which might be achieved simply by 
moving a vertical barrier at x = with the prescribed particle velo- 
city) instead of by changing the surface elevation rj the same typical 
differences will result if u is in the first case decreased from zero 
through negative values, and in the other case is gradually increased 



306 WATER WAVES 

so that it becomes positive (i.e. if the particles at x = are given in 
the first case a negative acceleration and in the second case a positive 
acceleration.) The qualitative difference between the two cases from 
the physical point of view is of course that in the first case it is a 
depression in the water surface and in the second case an elevation 
above the undisturbed surface — sometimes referred to later on as a 
hump— which propagates into still water. 

If we were to consider waves of very small amplitude so that we might 
linearize our equations (as was done in deriving equation (10.1.13)) 
there would be no essential qualitative distinction between the motions 
in the two cases; that there is actually a distinction between the two 
is a consequence of the nonlinearity of the differential equations. 

In the preceding section we have seen that the motions in either of 
our two cases can be described in the x, 2-plane by means of a family of 
straight characteristics which issue from the 2-axis. In Figure 10.4.1 
we show these characteristics together with a curve indicating a set 
of prescribed values for c = Vg(h-\-n) = c(t) at x = 0, which in 
turn result from prescribed values of r\ at that point. We assume that 
u = u = in the zone of quiet/. Hence the slope dx/dt of any straight 
characteristic issuing from a point t = r on the £-axis is given, in ac- 
cordance with (10.3.6) by 

dx 

(10.4.1) — = Sc(r)—2c . 

dt 

When t is varied (10.4.1 ) yields the complete set of straight characteris- 
tics in the zone //. The values of u and c along the same characteristic 
are constant (as we have seen in the preceding section), so that the 
value of u along a characteristic is determined, from (10.3.5) by 

(10.4.2) m(t) = 2[c(t) - c ], 
since u is assumed to be zero and c(r) is given. 

We are now in a position to note a crucial difference between the 
two cases described above. In the first of the two cases — i.e. that of a 
depression moving into still water— the elevation rj(t) at x = is 
assumed to be a decreasing function of t so that c(t) also decreases 
with increase of t. It follows that the slopes dx/dt of the straight line 
characteristics as given by (10.4.1) decrease as t increases* so that the 
family of straight characteristics diverge on moving out from the 

* One should observe that decreasing values of dx/dt mean that the charac- 
teristics make increasing angles with the a;-axis, i.e. that they become steeper with 
respect to the horizontal. 



LONG WAVES IN SHALLOW WATER 



30', 



/-axis. (This is the case indicated in Fig. 10.4.1). In the second case, 
however, the value of r\ and thus of c is assumed to be an increasing 
function of t at x = so that the straight characteristics must even- 



>lg(h + 77) 




Fig. 10.4.1. Propagation of pulses into still water 



c(0,t) 



tually intersect — in fact, they will have an envelope in general— and 
this in turn means that our problem can not be expected to have a 
continuous solution for values of x and t beyond those for which 
such intersections exist. In the first case the motion is continuous 
throughout. What happens in the second case beyond the point 
where the solution is continuous can not be discussed mathematically 
until we have widened our basic theory, but in terms of the physical 
behavior of the water we might expect the wave to break, or to devel- 
op what is called a bore,* some time after the solution ceases to be 
continuous. In later sections we propose to discuss the question of the 
development of breakers and bores in some detail. 

The two cases discussed above are the exact analogues of two cases 
well known in gas dynamics: Consider a long tube filled with gas at 
rest and closed by a piston at one section. If the piston is moved away 
from the gas with increasing speed in such a way as to cause a 
rarefaction wave to move into the quiet gas, then a continuous motion 
results. However, if the piston is pushed with increasing speed into 
the gas so as to create a compression wave, then such a wave always 

* In certain estuaries in various parts of the world the incoming tides from the 
ocean are sometimes observed to result in the formation of a nearly vertical wall 
of water, called a bore, which advances more or less unaltered in form over 
quite large distances. What is called a hydraulic jump is another phenomenon 
of the same sort. Such phenomena will be discussed in detail later on. 



308 



WATER WAVES 



develops eventually into a shock wave. That is, the development of 
a shock in gas dynamics is analogous to the development of a bore 
(and also of a hydraulic jump) in water. 



10.5. Propagation of depression waves into still water of constant depth 

In this section we give a detailed treatment of the first type of 
motion in which a depression of the water surface propagates into 
still water. However, it is interesting and instructive to prescribe the 
disturbance in terms of the velocity of the water rather than in terms 
of the surface elevation. We assume, in addition, that the velocity is 
prescribed by giving the displacement x = x(t) of the water particles 
originally in the vertical plane at x = 0,* and this, as we have remark- 
ed before, could be achieved experimentally simply by moving a ver- 
tical plate at the end of a tank in such a way that its displacement 
is x(t).** Figure 10.5.1 indicates the straight characteristics which 




= const. 



Fig. 10.5.1. A depression wave 

initiate on the "piston curve" x = x(t). The piston is assumed to 
start from rest and move in the negative direction with increasing 
speed until it reaches a certain speed w < 0, after which the speed 
remains constant. That is, x t decreases monotonically from zero at 
t = until it attains the value w, after which it stays constant at that 
value. In Fig. 10.5.1 this point is marked B; clearly the piston curve is 

* In our theory, it should be recalled, the particles originally in a vertical 
plane remain always in a vertical plane. 

** Moving such a plate at the end of a tank of course corresponds in gas dyna- 
mics to moving a piston in a gas-filled tube. 



LONG WAVES IN SHALLOW WATER 309 

a straight line from there one. At any point A on the piston curve we 
have u A — x t {t), corresponding to the physical assumption that the 
water particles in contact with the piston remain in contact with it 
and thus have the same velocity. If we consider the curved character- 
istic drawn from A back to the initial characteristic Cj which termina- 
tes the zone / of rest we obtain from (10.3.5) the relation 

(10.5.1) c A = \u A + c , 

since in our case u = 0. The slope of the straight characteristic at A 
is thus given by (cf. (10.4.1)): 

dx 3 

(10.5.2) — = -u A +c . 

Since we have assumed that u A = x t (t) always decreases as t increases 
until x t = w it follows from (10.5.2) that dxjdt also decreases as t in- 
creases in this range of values of t so that the characteristics diverge 
as they go outward from the piston curve. Beyond the point B the 
straight characteristics are parallel straight lines, since u A — w = 
const, on that part of the piston curve, and the state of the water is 
therefore constant in the zone marked III in Fig. 10.5.1. The zone II 
is thus a region of non-constant state connecting two regions of differ- 
ent constant states. Since c A = Vg(h-\-r) A ), where r\ A refers to the 
elevation of the water surface at the piston, it follows from (10.5.1) 
that r\ A decreases in the zone 77 as t increases, i.e. the water surface at 
the "piston" moves downward as the piston moves to the left, since we 
assume that u A decreases as A moves out along the piston curve. Since 
u and c are constant along any straight characteristic it is not difficult 
to describe the character of the motion corresponding to the disturbed 
zone II at any time t: Consider any straight line t = const. Its inter- 
section with a characteristic yields the values of u and c at that point 
—they are the values of u and c which are attached to that character- 
istic. Since the characteristics diverge from the piston curve one sees 
that the elevation r\ steadily increases upon moving from the piston to 
the right and the particle velocity decreases in magnitude, until the 
initial characteristic Cj is reached after which the water is undisturbed. 
On the other hand, if attention is fixed on a definite point x > in the 
water and the motion is observed as the time increases it is clear— once 
more because the characteristics diverge— that the water remains un- 
disturbed until the time reaches the value determined by x = c t, after 
which the water surface falls steadily while the water particles passing 



310 



WATER WAVES 



that point move more and more rapidly in the negative ^-direction. 
In the foregoing discussion of a depression we have made an assump- 
tion without saying so explicitly, i.e. that the speed u A of the piston is 
such that c A = \u A -f- c (cf. (10.5.1)) is not negative, and this in 
turn requires that 

(10.5.3) — u A ^ 2c . 

Since — u A increases monotonically to the terminal value — w it 
follows that — w must be assumed in the above discussion to have at 




Fig. 10.5.2. A limit case 



most the value 2c . The limit case in which — w just equals 2c is 
interesting. Since the straight characteristics have the slope dx/dt = 
u + c and since c A = from (10.5.1) when u A = — 2c , it follows in 
this case that dx/dt = u A on the straight part of the piston curve. 
But this means that the straight characteristics have all coalesced 
into the piston curve itself in this region, or in other words that the 
zone 277 has disappeared in this limit case. The circumstances are 
indicated in Fig. 10.5.2. At the front of the wave for values of x to the 
left of B the elevation r\ A of the water is equal to — h from c A = 
Vg(h -f i?a) — 0> which means that the water surface just touches the 
bottom at the advancing front of the wave. 

It is now clear what would happen if the terminal speed — w of the 
piston were greater than 2c : The zone II would terminate on the 
tangent to the piston curve drawn from the point where the piston 
speed — x t just equals 2c . The region between this terminal charac- 
teristic and the remainder of the piston curve beyond it might be 
called the zone of cavitation, since no water would exist for (x, t) 



LONG WAVES IN SHALLOW WATER 



311 



values in such a region. In other words, the piston eventually pulls 
itself completely free from the water in this case. Quite generally we 
see that the piston will lose contact with the water (under the cir- 
cumstances postulated in this section, of course) if, and only if, it 
finally exceeds the speed 2c . Once this happens it is clear that the 
piston has no further effect on the motion of the water. These circum- 
stances are indicated in Fig. 10.5.3. 




Fig. 10.5.3. Case of cavitation 

If the acceleration of the piston is assumed to be infinite so that 
its speed changes instantly from zero to the constant terminal value 
— w, the motion which results can be described very simply by ex- 
plicit formulas. The general situation in the x, Z-plane is indicated in 
Fig. 10.5.4. This case might be considered a limit case of the one 
indicated in Fig. 10.5.1 which results when the portion of the piston 
curve extending from the origin to point B shrinks to a point. The 



•*■(¥ + e J» 




Fig. 10.5.4. Centered simple wave 



312 WATER WAVES 

consequence is that the straight characteristics in zone II all pass 
through the origin. The zone III is again one of constant state. In 
the zone II we have obviously for the slopes of the characteristics 

dx x 

(10.5.4) — = — . 
K ' dt t 

At the same time we have from (10.5.2) dx/dt = § u -f c so that 

x 3 

(10.5.5) - = -u+c Q . 

It follows that the zone II is terminated on the upper side by the line 

(10.5.6) x = I- zv + c W 

From (10.5.5) and (10.5.1) we can obtain the values of u and c within 
zone 17: 

2 /x \ 

(10.5.7) U = J\i~ C °) and 



(10.5.8) c = ju+c 



Mt +2 4 



Since c^Owe must have — x/t ^ 2c so that — w must be ^ 2c 

from (10.5.6) in conformity with a similar result above. If w = — 2c , 

the terminal characteristic of zone II is given, from (10.5.6), by 

x = — 2c t = wt and this line falls on the piston curve since the slope 

of the piston curve is w. In this limit case, therefore, the zone /// 

collapses into the piston curve. If the piston is moved at still higher 

speed, then cavitation occurs as in the cases discussed above since 

c = at the front of the wave, or in other words, the water surface 

touches the bottom. 

From (10.5.8) we can calculate the elevation r\ of the water surface 

since c = Vg(h + rj) ; 

1 lx \ 2 

(10.5.9) n +h = —{j + 2c j . 

In the case of incipient cavitation, i.e. — w = 2c , we have rj = — h 
at the front of the wave. The curve of the water surface at any time t 
is a parabola from the front of the wave to the point x = c t (cor- 
responding to the characteristic which delimits the zone of quiet), 
after which it is horizontal. In Fig. 10.5.5 the total depth rj + h of 



LONG WAVES IN SHALLOW WATER 



318 



the water is plotted against x for a fixed time t. The surface of the 
water is tangent to the bottom at the front x = — 2c t of the moving 
water. The region in which the water is in motion extends from this 
point back to the point x = c t. From (10.5.7) we can draw the follow- 
ing somewhat unexpected conclusion in this case: Since t may be 
given arbitrarily large values it follows that the velocity u of the water 
at any fixed point x tends to the values — § c as t grows large. 
The case of cavitation may have a certain interest in practice: the 
motion of the water might be considered as an approximation to the 
flow which would result from the sudden destruction of a dam built 
in a valley with very steep sides and not too great bottom slope (cf. 



A77+h 




Fig. 10.5.5. Breaking of a dam 

the paper of Re [R.5]). If the water behind the dam were 200 feet 
high, for example, our results indicate that the front of the wave 
would move down the valley at a speed of about 110 miles per hour. 
By setting x = in (10.5.9) we observe that the depth of the water 
at the site of the dam is always constant and has the value pi, 
i.e. four-ninths of the original depth of the water behind the dam. 
The velocity of the water at this point is also constant and has the 
value u = — |c = — § Vgh, as we see from (10.5.7). The volume 
rate of discharge of water at the original location of the dam is thus 
constant. 

So far we have not considered the motion of the individual water 
particles. However, that is readily done in all cases once the velocity 
u(x, t) is known: We have only to integrate the ordinary differential 
equation 

dx 
(10.5.10) 



dt 



u. 



In zone 77 in our present case we have 



314 WATER WAVES 

dx 2 Ix \ 

< 10 - 5 - n > * = «(t-4 

By setting £ = x + 2c 2 one finds readily that | satisfies the differen- 
tial equation d£/eft = 2£/32, from which £ = ^ 2/3 with ^4 an arbitrary 
constant. Hence we have for the position x(t) of any particle in zone 77 

(10.5.12) x = t{At~ llz - 2c }. 

In the case of cavitation this formula holds for arbitrarily large t so 
that we have for large t the asymptotic expression for x: 

(10.5.13) x~ — 2c t. 

(This is not in contradiction with our above result that u~ — fc 
for large t and fixed x since in that case different particles pass the 
point in question at different times, while (10.5.13) refers always to 
the same particle). 

In the first section of Chapter 12 this same problem of the breaking 
of a dam will be treated by using the exact nonlinear theory in such 
a manner as to determine the motion during its early stages after the 
dam has been broken — in other words, at the times when the shallow 
water theory is most likely to be inaccurate. 

10.6. Discontinuity, or shock, conditions 

The difference in behavior of a depression which propagates into 
still water as compared with the behavior of a hump has already been 
pointed out: in the first case the motion is continuous throughout, but 
in the second case the motion can not be continuous after a certain 
time. The general situation is indicated in Fig. 10.6.1, which shows 
the characteristics in the x, /-plane for the motion which results when 
a "piston" at the end of a tank is pushed into the water with steadily 
increased speed. As before, the slope dx/dt of a straight characteristic 
issuing from the "piston curve" x = x(t) is given (cf. (10.5.2)) by 
dx/dt = ^u A + c o> m which u A = x t (t) is the velocity of the piston. 
Since u A is assumed to increase with t it is clear that the characteris- 
tics will cut each other. In general, they have an envelope as indi- 
cated by the heavy line in the figure. The continuous solutions 
furnished by our theory, which have been the only ones under con- 
sideration so far, are thus valid in the region of the x, /-plane between 
the initial characteristic and the piston curve up to the curved 
characteristic (indicated by the curve segment ED) through the 



LONG WAVES IN SHALLOW WATER 



315 



"first" point E on the envelope of the straight characteristics, but 
not beyond ED. 

What happens "beyond the envelope" can in principle therefore 




Fig. 10.6.1. Initial point of breaking 

not be studied by the theory presented up to now. However, it seems 
very likely that discontinuous solutions may develop as the time in- 
creases beyond the value corresponding to the point E, which are 
then to be interpreted physically as motions involving the gradual 
development of bores and breakers in the water. 



1 


i; + h 














I 


i 








1 
1 
1 
1 
1 
1 


77+h 





o (t) 



ett) 



x=a,(t) 



Fig. 10.6.2. Discontinuity conditions 

There is a particularly simple limit case of the situation indicated 
in Fig. 10.6.2 for which a discontinuous solution can be found once 
we have obtained the discontinuity conditions that result from the 



316 WATER WAVES 

fundamental laws of mechanics. That is the case in which the "piston" 
is accelerated instantaneously from rest to a constant forward velocity 
so that the piston curve is a straight line issuing from the origin in 
the x, 2-plane. It is the exact counterpart of the case discussed at the 
end of the preceding section in which the piston was withdrawn from 
the water at a uniform speed. 

To obtain the conditions at a discontinuity we consider a region 
made up of the water lying between two vertical planes x = a (t) 
and x — %(0 with a ± > a and such that these planes contain always 
the same particles. Such an assumption can be made, we recall from 
Chapter 2, since in our theory the particles which are in a vertical 
plane at any instant always remain in a vertical plane. Hence the 
horizontal particle velocity component u is the same throughout any 
vertical plane. We now suppose that there is a finite discontinuity in 
the surface elevation r\ at a point x = £(t) within the column of water 
between x = a (t) and x = a x {t), as indicated in Fig. 10.6.2. 

The laws of conservation of mass and of momentum as applied to 
our column of water yield the relations 

d r aiit) 

(10.6.1) — Q{r]+h)dx = 

and 

h (t) 



d f°iW 



g(r] + h)u dx = 


Po d y - Vi dy 

—h J —h 


= iSQiVo + hf - lgQ( Vl + h)\ 


ia v = SQin — y 


) for the pressure in the water is 



(10.6.2) dt 



when the formula 
used. The second relation states that the change in momentum of the 
water column is equal to the difference of the resultant forces over the 
end sections of the column. 

The integrals in these relations have the form 

r>a x (t) 
I = yj(x } t) dx 

Ja (t) 

in which ip(x, t) has a discontinuity at x = £(/). Differentiation of this 
integral yields the relation 

h (t) 

yj dx 



di d r*(*> d 

dt = dt} ao{t) y)dx ~ { "dt 



1(0 



(10.6.3) f«i(*>tty 

' ax -f- 

y)( ai (t), t)u x - y>(£ + , t)£(t). 



( ~dx + ^ (£-> 0i(0 - y>Mt)> t)u 

Ja (t) & 



LONG WAVES IN SHALLOW WATER 317 

The quantities u and u ± are the velocities a (t) and a ± (t) at the ends 
of the column, £ is the velocity of the discontinuity, and ip(£_, t) and 
y)(£+, t) mean that the limit values of tp to the left and to the right of 
x = | respectively are to be taken. We wish to consider the limit case 
in which the length of the column tends to zero in such a way that 
the discontinuity remains inside the column. When we do so the 
integral on the right-hand side of (10.6.3) tends to zero and we obtain 

dl 

(10.6.4) lim — = Vi^i - Wo 

in which v 1 and v are the relative velocities given by 

>i = «*i - & 



(10.6.5 

' ^o = u o - I 

and ip x and yi Q refer to the limit values of tp to the right and to the left 
of the discontinuity, respectively. The important quantities v and v 1 
are obviously the flow velocities relative to the moving discontinuity. 
Upon making use of (10.6.4) and (10.6.5) for the limit cases which 
arise from (10.6.1) and (10.6.2) we obtain the following conditions 

(10.6.6) Q ( m + h)v x - q( Vo + h)v = 
and 

(10.6.7) rt?? 1 +/iM-^(j}o+^)Vo-kho+^) 2 -k( ? ?i+ /i ) 2 - 

If we introduce, as in section 10.1, the quantities q and p (which are 
the analogues of the density and pressure in gas dynamics) by the 
relations (cf. (10.1.3) and (10.1.4)) 

(10.6.8) Q = Q(r]+h) 
and 

(10.6.9) p = ^L(r,+hY= fg°, 

we obtain in place of (10.6.6) and (10.6.7) the discontinuity conditions 

(10.6.10) q iVi - o v , 
and 

(10.6.11) QiU 1 v 1 — Q u v = p — p v 

The last two relations are identical in form with the mechanical con- 
ditions for a shock wave in gas dynamics when the latter are expressed 
in terms of velocity, density and pressure changes. 



318 WATER WAVES 

Henceforth we shall often refer to a discontinuity satisfying (10.6. 
10) and (10.6.11) as a shock wave or simply as a shock even though 
such an occurrence is better known in fluid mechanics as a bore, or 
if it is stationary as a hydraulic jump. 

Since u x — u = v x — v from (10.6.5) it is easily seen that the 
shock conditions (10.6.10) and (10.6.11) can be written in the form 

(10.6.12) { -^ = W» = ™> 

in which m represents the mass flux across the shock front. 

To fix the motion on both sides of the shock five quantities are 
needed; i.e. the particle velocities u , u v the elevations r) and r\ x (or, 
what is the same, the "pressures" p or the "densities" q as given by 
(10.6.8) and (10.6.9) on both sides of the shock), and the velocity £ 
of the shock. Evidently the relative velocities v and v 1 would then 
be determined. Since the five quantities satisfy the two relations 
(10.6.12) we see that in general only three of the five quantities could 
be prescribed arbitrarily. Since the equations to be satisfied are not 
linear it is not a priori clear whether solutions can be found for two 
of the quantities when any other three are arbitrarily prescribed or 
whether such solutions would be unique. We want to investigate this 
question in a number of important special cases. 

Before doing so, however, it is important to consider the energy 
balance across a shock. The fact is, as we shall see shortly, that the 
law of conservation of energy does not hold across a shock, but rather 
the particles crossing* the shock must either lose or gain in energy. 
Since we do not wish to postulate the existence of energy sources at 
the shock front capable of increasing the energy of the water particles 
as they pass through it, we assume from now on that the water 
particles do not gain energy upon crossing a shock front. This will in 
effect furnish us with an inequality which in conjunction with the 
two shock relations (10.6.12) leads in all of our cases to unique solu- 
tions of the physical problems. We turn, then, to a consideration of 
the energy balance across a shock, which we can easily do by following 

* It is important to observe that the water particles always do cross a shock 
front: the quantity m in (10.6.12), the mass flux through the shock front, is 
different from zero if there is an actual discontinuity since otherwise v x = v == 0, 
u x = u = |, and p = p x and hence Q = ^ — in other words the motion is 
continuous. There is thus no analogue in our theory of what is called a contact 
discontinuity in gas dynamics in which velocity and pressure are continuous, 
but the density and temperature may be discontinuous. 



LONG WAVES IN SHALLOW WATER 319 

the same procedure that was used to derive the shock relations (10.6. 
10) and (10.6.11). For the rate of change dE/dt of the energy E in the 
water column of Fig. 10.6.2 we have, as one can readily verify: 

dE d I f a iW r u 2 go 

(10.6.13) * M Ua o«) L 2 2 



dx 



+ \\ PA dy - J* p u dy 



and this in turn yields in the limit when a -» a v through use of 

(10.6.5), (10.6.8), (10.6.9), and the hydrostatic pressure law, the 

relation 

dE 9 _ 9 

(10.6.14) — = \q 1 u\v l - Iq uIv q + y 1 v 1 - p v + p ± u ± - p u 

for the rate at which energy is created or destroyed at the shock front. 
If we multiply (10.6.11) by £ on both sides and then subtract from 

(10.6.14) the result is an equation which can be written after some 
manipulation and use of (10.6.5) in the form 

dE 

(10.6.15) — = m {i(vf - vl) + 2(pjQ 1 - p 0i Iq )} 

in which m is the mass flux through the shock front defined in 
(10.6.12). In this way we express dE/dt entirely in terms of the 
relative velocities v and v x and the change in depth. By eliminating 
v x and v through use of v x = m/Q 1 and v = ml~Q and replacing p x and 
p in terms of q 1 and £ we can express dE/dt in terms of £ and g x ; 
the result is readily found to be expressible in the simple form 

dE mg (p — Pi) 3 

(10.6.16) —-= _ ^°_ y i > , 

at q 4^0 

We see therefore that energy is not conserved unless q Q = q v i.e. unless 
the motion is continuous. Since Q — Qi = Qi^o ~ Vi) ^ follows from 
(10.6.16) that the rate of change of the energy of the particles crossing 
the shock is proportional to the cube of the difference in the depth of 
the water on the two sides of the shock, or as we could also put it in 
case tj — rj 1 is considered to be a small quantity: the rate of change 
of energy is of third order in the "jump" of elevation of the water 
surface. 

The statement that the law of conservation of energy does not hold 
in the case of a bore in water must be taken cum grano salis. What we 
mean is of course that the energy balance can not be maintained 



320 



WATER WAVES 



through the sole action of the mechanical forces postulated in the 
above theory. The results of our theory of the bore and the hydraulic 
jump are therefore to be interpreted as an idealization of the actual 
occurrences in which the losses in mechanical energy are accounted 
for through the production of heat due to turbulence at the front of 
the shock (cf. the photograph of the bore in the Tsien-Tang river 
shown in Fig. 10.6.3). In compressible gas dynamics the theory used 




Fig. 10.6.3. Bore in the Tsien-Tang River 



allows for the conversion of mechanical energy into heat so that the 
law of conservation of energy holds across a shock in that theory. 
The analogue of the loss in mechanical energy across a shock in water 
is the increase in entropy across a shock in gas dynamics; furthermore, 
both of these discontinuous changes are of third order in the differen- 
ces of "density" on the two sides of the shock. 

We have tacitly chosen as the positive direction of the a?-axis, and 
hence of all velocities, the direction from the side toward the side 1 
(cf. Fig. 10.6.2). Suppose now that the mass flux m is assumed to be 
positive; it follows from (10.6.12) and the fact that g and g t are posi- 
tive that v and v 1 are also positive and hence that the water particles 
cross the shock front in the direction from the side toward the side 1. 
Our condition that the water particles can not gain in energy on cross- 
ing the shock then requires, as we see at once from (10.6.16) since m, 



LONG WAVES IN SHALLOW WATER 321 

g, q, £> , and Q ± are all positive, that q < q v In other words, our energy 
condition requires that the particles always move across the shock from 
a region of lower total depth to one of higher total depth.* Since the mass 
flux m is not zero unless the flow is continuous, and hence there is no 
shock, it is possible to define uniquely the two sides of the shock by the 
following useful convention: the front and back sides of the shock are 
distinguished by the fact that the mass flux passes through the shock 
from front to back, or, as one could also put it, the water crosses the 
shock from the front side toward the back side. Our conclusion based 
on the assumed loss of energy across the shock can be interpreted in 
terms of this convention as follows: the water level is always lower on 
the front side of the shock than on the back side. 

For the further discussion of the shock relations it is important to 
observe that all of them, including the relation (10.6.16) for the 
energy loss, can be written in such a way as to involve only the velocities 
v Q and v Y of the water particles relative to the shock front and not the abso- 
lute velocities u and u v It follows that we may always assume one of 
the three velocities u , u v £ to be zero if we wish, with no essential loss 
of generality, because the laws of mechanics are in any case invariant 
with respect to axes moving with constant velocity, and adding the 
same constant to u , u x and £ does not affect the values of v and v x . 

Let us assume then that u = 0, i.e. that the water is at rest on one 
side of the shock. Also, we write the second of the shock conditions 
(10.6.12) in the form 

(10.6.17) Vl v = V ~ — V } , 

Qo — Qi 

which follows from mv 1 = QqVqVj^ and mv = q^^q and (10.6.12). 
From u = we have v = — £ and v x = u x — £ (cf. (10.6.5)) so that 
(10.6.17) takes the form 

(10-6.18) -£K-£)=^fe> + £i) 

upon making use of p = gg 2 i'2Q (cf. (10.6.9)). The first shock condition 
now takes the form 

(10-6.19) ^K - £) = - £ £, 

so that (10.6.18) can be written 

(10.6.20) £2 =^Ml +1^1 

2 £ \ go/ 



* 



This conclusion was first stated by Rayleigh [R.3J. 



322 WATER WAVES 

if u x is eliminated, or it may be written in the form 

do.6.21) -iK-i^t^-"-^) 

if q is eliminated. Thus (10.6.19) together with either (10.6.20) or 
(10.6.21) are ways of expressing the shock conditions when u = 0. 

We are now in a position to discuss some important special cases. 
Having fixed the value of u , i.e. u = 0, at most two of the remaining 
quantities £, q , q v and u x can be prescribed arbitrarily. For our later 
discussion it is useful to single out the following two cases: Case 1. p x 
and ~q are given, i.e. the depth of the water on both sides of the shock 
and the velocity on one side are given. Case 2. q x and u x are given, i.e. 
the velocity of the water on both sides of the shock and the depth of 
the water on one side are given. We proceed to discuss these cases in 
detail. 

Case 1. From (10.6.20) we see that £ 2 is determined for any arbitrary 
values (positive, of course) of q and q v i.e. of the water depths. Hence 
£ is determined by (10.6.20) only within sign. Suppose now that 
Q! > g . In this case the side is, as we have seen above, the front 
side of the shock, and since u = the shock front must move in the 
direction from the side 1 toward the side in order that the mass flux 
should pass through the shock from front to back. 

Hence if it is once decided whether the side is to the left or to the 
right of the side 1 the sign of | is uniquely fixed. If, as in Fig. 10.6.4, 



e «- 



u, <0 



x 
Fig. 10.6.4. Bore advancing into still water 

the side is chosen to the left of the side 1, and the a?-direction is posi- 
tive to the right, it follows that £ is negative, as indicated. It is useful 
to introduce the depths h and h ± of the water on the two sides of the 
shock: 

(10.6.22) [.*. = *+%. 



LONG WAVES IN SHALLOW WATER 323 

and to express (10.6.20) in terms of these quantities. The result for £ 
in our case is 

(10.6.23) , = _|/ g _(___) 

as one readily sees from J> t = gh^ From (10.6.23) we draw the im- 
portant conclusion: Since h ± > h , the shock speed | £ | is greater than 
Vgh since h < (h ± + h )/2 < h v Also, in the case u = we have 
from (10.6.19) 



(10.6.24) % = i 



-» 



so that //ie velocity of the water behind the shock has the same sign as 
£ (since h /h x < 1 ) but is less than £ numerically. 

Finally, it is very important to consider the speed v x of the shock 
front relative to the water particles behind it: from (10.6.24) we have 

(10.6.25) Vl = u x — £ = — t^£, 

h ± 

and this in turn can be expressed through use of (10.6.23) in the 
form 



(10.6.26) 



]/ h ( K + h \ 



so that v x < Vgh v In other words, the speed of the shock relative to 
the water particles behind the shock is less than the wave propagation 
speed Vgh x in the water behind the shock. Hence a small disturbance 
created behind a shock will eventually catch up with it. Although the 
conclusion was drawn for the special case u = it holds quite 
generally for the shock velocities relative to the motion of the water 
on both sides of a shock, in view of earlier remarks on the dependence 
of the shock relations on these relative velocities. 

The case illustrated by Fig. 10.6.4 is that of a shock advancing into 
still water. The fact that £ is in this case of necessity negative is a 
consequence of the assumption of an energy loss across the shock. It 
is worth while to restate this conclusion in the negative sense, as 
follows: a depression shock can not exist, i.e. a shock wave which 
leaves still water at reduced depth behind it should not be observed 
in nature.* The observations bear out this conclusion. Bores advancing 

* In gas dynamics the analogous situation occurs: only compression shocks 
and not rarefaction shocks can exist. 



324 



WATER WAVES 




Fig. 10.6.5. Reflection from a rigid wall 



Z--0 



Fig. 10.6.6. Hydraulic jump 



into still water are well known, but depression waves are always 
smooth. 

Instead of assuming that g ± > jo (or that h x > h ) as in the case of 
Fig. 10.6.4 we may assume q x < g (or h x < h Q ), so that the side 1 is 
the front side. In other words the water is at rest on the back side of 
the shock in this case. If the front side is taken on the right, the situa- 
tion is as indicated in Fig. 10.6.5. In this case £ must be positive 
and u x negative in order that the mass flux should take place from the 
side 1 to the side 0. The value of u x is given by (10.6.24) in this case 
also. The case of Fig. 10.6.5 might be realized in practice as the result 
of reflection of a stream of water from a rigid wall so that the water in 
contact with the wall is brought to rest. We shall return to this case 
later. 

In the above two cases we considered u to be zero. However, we 
know that we may add any constant velocity to the whole system 
without invalidating the shock conditions. It is of interest to consider 
the motion which arises when the velocity — | is added to u , u v and 
£ in the case shown in Fig. 10.6.4. The result is the motion indicated 
by Fig. 10.6.6 in which the shock front is stationary. This case — one 
of frequent occurrence in nature — is commonly referred to as the 
hydraulic jump. From our preceding discussion we see that the water 
always moves from the side of lower elevation to the side of higher 
elevation. The velocities u and % are both positive, and u Q > u v 



LONG WAVES IN SHALLOW WATER 



325 



Also the velocity u on the incoming side is greater than the wave 
propagation speed Vgh on that side while the velocity u x is less than 
Vgh v This follows at once from the known facts concerning the re- 
lative shock velocities and the fact that u Q and u x are the velocities 
relative to the shock front in this case. The hydraulic engineers refer 
to this as a transition from supercritical to subcritical speed. 

Case 2. We recall that in this case u = 0, u v and q ± (or h x ) are 
assumed given and £ and h are to be determined. The value of £ is 
to be determined from (10.6.21). To study this relation it is conve- 
nient to set x = — £ and y = u x — £ so that (10.6.21 ) can be replaced 

by 

( y = k 2 x/(x 2 - k 2 ), k 2 = — 
(10.6.27) * /V h 2 

\ y = u x + x. 

In Fig. 10.6.7 we have indicated these two curves, whose intersections 
yield the solutions £ = — x of (10.6.21). The first equation is re- 
presented by a curve with three branches having two asymptotes 
x = ± k. As one sees readily, there are always three different real 
roots for — £ no matter what values are chosen for the positive quan- 
tity k 2 = g/^/2 and for the velocity u x . Furthermore, one root £ + = 
— x_ is always positive, another £_ = — x + is negative, while the 
third £ = — x lies between the other two. However, the third root 
£ = — x must be rejected because it is not compatible with (10.6.19): 
Since ^ and £> are both positive it follows that x = — £ and y = 




P'ig. 10.6.7. Graphical solution of shock conditions 



326 WATER WAVES 

u x — | must have the same sign. But the sign of y = y corresponding 
to x = x is always the negative of x as one sees from Fig. 10.6.7. 
(If u x = ,then x = y = 0, but there is no shock discontinuity in 
this case.) The other two roots, however, are such that the signs of 
— £ and u x — £ are the same. In the case 2, therefore, equation 
(10.6.21) furnishes two different values of £ which have opposite 
signs and these values when inserted in (10.6.19) furnish two values 
of the depth h . The two cases are again those illustrated in Figs. 
10.6.4 and 10.6.5. An appropriate choice of one of the two roots must 
be made in accordance with the given physical situation, as will be 
illustrated in one of the problems to be discussed in the next section. 
Before proceeding to the detailed discussion of special problems 
involving shocks it is perhaps worth while to sum up briefly the main 
facts derived in this section concerning them: the five essential quan- 
tities defining a shock wave— £, u , u ± ; @ , q x (or, what is the same, 
h and h^— must satisfy the shock conditions (10.6.12). If it is as- 
sumed in addition that the water particles may lose energy on crossing 
the shock but not gain it, then it is found that the shock wave travels 
always in such a direction that the water particles crossing it pass from 
the side of lower depth to the side of higher depth. If h Q < h v so that the 
side is the front side of the shock, the speeds | v | and \v x \ of the zvater 
relative to the shock front satisfy the inequalities 

(10.6.28) ' °' tl' 

\\v x \< Vgh v 

In hydraulics it is customary to say that the velocity relative to the 
shock is supercritical on the front side (i.e. greater than the wave 
propagation speed corresponding to the water depth on that side) 
and subcritical on the back side of the shock.* 



10.7. Constant shocks: bore, hydraulic jump, reflection from a 
rigid wall 

In the preceding section shock discontinuities were studied for the 
purpose of obtaining the relations which must hold on the two sides 
of the shock, and nothing was specified about the motion otherwise 
except that the shock under discussion should be the only disconti- 

* In gas dynamics the analogous inequalities lead to the statement that the 
flow velocity relative to the shock front is supersonic with respect to the gas 
on the side of lower density and subsonic with respect to the gas on the other side. 



LONG WAVES IN SHALLOW WATER 327 

nuity in a small portion of the fluid on both sides of it. In the present 
and following sections we wish to consider motions which are conti- 
nuous except for the occurrence of a single shock. Furthermore we 
shall limit our investigations in this section to cases in which the mo- 
tion on each of the two sides of the shock has constant velocity and 
depth. These motions, or flows, are evidently of a very special cha- 
racter, but they are easy to describe and also of frequent occurrence 
in nature. 

It is perhaps not without interest in this connection to observe that 
the only steady and continuous wave motions (i.e., motions in which 
the velocity u and wave propagation speed c = Vg(h + rj) are in- 
dependent of the time) furnished by our theory for the case of constant 
depth h are the constant states u = const., c = const. This follows 
from the original dynamical equations (10.1.8) and (10.1.9). In fact, 
when u and c are assumed to be functions of x alone these equa- 
tions reduce to 

du dc 

u- h 2c — = 0, and 

dx dx 

dc du 

2u — + c — = 
dx dx 

for the case in which h = const, (and so H x = 0). These equations are 
immediately integrable to yield u 2 + 2c 2 == const, and uc 2 = const, 
and these two relations are simultaneously satisfied only for constant 
values of u and c. On the other hand, any constant values whatever 
could be taken for u and c. The cases we discuss in this section are 
motions which result by piecing together two such steady motions 
(each with a different constant value for the depth and velocity) 
through a shock which moves with constant velocity. In this case the 
motion as a whole would be steady if observed from a coordinate 
system attached to the moving shock front. In view of our above dis- 
cussion it is clear that such a motion with a single shock discontinuity 
is the most general progressing wave which propagates unchanged in 
form that could be obtained from our theory.* 

Let us consider now the problem referred to at the beginning of the 
preceding section: a vertical plate — or piston, as we have called it — at 

* This result should not be taken to mean that the so-called "solitary wave" 
does not exist. (By a solitary wave is meant a continuous wave in the form of a 
single elevation which propagates unaltered in form.) It means only that our 
approximate theory is not accurate enough to furnish such a solitary wave. This 
is a point which will be discussed more fully in section 10.9. 



328 WATER WAVES 

the left end of a tank full of water at rest is suddenly pushed into the 
water at constant velocity w. As we could infer from the discussion 
at the beginning of the preceding section the motion must be dis- 
continuous from the very beginning— or, as we could also put it, the 
"first" point on the envelope of the characteristics would occur at 
t = 0. Since the piston moves with constant velocity we might expect 
the resulting motion to be a shock wave advancing into the still water 
and leaving a constant state behind such that the water particles move 
with the piston velocity w. The circumstances for such an assumed 
motion are indicated in Fig. 10.7.1, which shows the x, 2-plane together 
with the water surface at a certain time t . We know that the constant 
states on each side of the shock satisfy our differential equations. In 
addition, we show that they can always be "connected" through a 
shock discontinuity which satisfies the shock relations derived in the 
preceding section. In fact, the relations (10.6.18) and (10.6.19) yield 
through elimination of p 2 = gh ± the relation 

(10.7.1) _| (K) _^ ) = ^(l__^ 1 ) 

for | in terms of w and the depth h in the still water, when we set 
Qo = qK- Equation (10.7.1) is the same as (10.6.21) except that q 
replaces q 19 and the discussion of its roots | follows exactly the same 
lines as for (10.6.21): for each h > and any w ^ the cubic equa- 
tion (10.7.1) has three roots for |: one negative, another positive, and 
a third which has a value between these two. In the present case the 
positive root for | must be taken in order to satisfy our energy con- 
dition (cf. the discussion based on (10.6.27) of the preceding section) 
since the side is the front side of the shock. Once | + has been calcu- 
lated from (10.7.1 ) we can determine the depth of the water h x behind 
the shock from the first shock condition 

(10.7.2) h t (w - i + ) = - h £ + . 

It is therefore clear that a motion of the sort indicated in Fig. 10.7.1 
can be determined in a way which is compatible with all of our con- 
ditions.* 

A few further remarks about the above motion are of interest. In 

* It should be pointed out that our discussion yields a discontinuous solution 
of the differential equations, but does not prove that it is the only one which 
might exist. However, it has been shown by Goldner [G.6] that our solution would 
be unique under rather general assumptions regarding the type of functions 
admitted as possible solutions. 



LONG WAVES IN SHALLOW WATER 



329 




Piston 



Fig. 10.7.1. A bore with constant speed and height 



I 

(2) gg 

I 

















c_ * 


u„=0 


h 2 










h, 


u .— * 














1 


1 






Fig. 10.7.2. Reflection of a bore from a rigid wall 



330 WATER WAVES 

Fig. 10.7.1 we have indicated the line x = c t } c = Vgh , which 
would be the initial characteristic terminating the state of rest if the 
motion were continuous, i.e. if the disturbance proceeded into still 
water with the wave speed c for water of the depth h . We know, 
however, from our discussion of the preceding section that the shock 
speed i is greater than c , which accounts for the position of the shock 
line x — gt below the line x = c t in Fig. 10.7.1. On the other hand 
we know that the velocity w — | of the water particles behind the 
shock relative to the shock is less than the wave speed c x = Vgh ± in 
the water on that side. It follows, therefore, that a new disturbance 
created in the water behind the shock should catch up with it since 
the front of such a disturbance would always move relative to the 
water particles with a velocity at least equal to c v For example, if the 
piston were to be decelerated at a certain moment a continuous de- 
pression wave would be created at the piston which would finally 
catch up with the shock front, and a complicated interaction process 
would then occur. 

The case we have treated above corresponds to the propagation of 
a bore into still water. If we were to superimpose a constant velocity 
— | on the water in the motion illustrated by Fig. 10.7.1 the result 
would be the motion called a hydraulic jump in which the shock front 
is stationary. We need not consider this case further. 

We treat next the problem of the reflection of a shock wave from 
a rigid vertical wall by following essentially the same procedure as 
above. The circumstances are shown in Fig. 10.7.2. We have an 
incoming shock moving toward the rigid wall from the left into still 
water of depth h . The shock is reflected from the wall leaving still 
water of depth h 2 behind it. Since the water in contact with the wall 
should be at rest, such an assumed motion is at least a plausible one. 
We proceed to show that the motion is compatible with our shock 
conditions and we calculate the height h 2 of the reflected wave. 

We assume that h ± and w 1 = w, the depth and the velocity of the 
water behind the shock, are known. The shock speed £ + is then de- 
termined by taking the largest of the three roots of the cubic equation 
(10.6.21), which we write down again in the form 



(10.7.3) - £(«, - |) 



fKi-*) 



Once | + has been determined, the depth h in front of the shock is 
fixed from the first shock condition, which is in the present case 



LONG WAVES IN SHALLOW WATER 



331 



(10.7.4) (» - l + )h ± = - l + h Q . 

To determine the reflected shock we may once more evidently make 



































4 
3 
2 








1 
/> 2 _ 6(^)-l 


>' = 1.4 
















* 




+ J>- 














/ 

/ . 
/ / 


' ^ 


















■JL 


X 























1 


1 


1 


1 


1 


1 


1 


1 


1 


1 





8 10 12 14 16 

(a) 



ho 


. 






















100 


- 
















y 






80 
60 
















/ 






























40 

20 
















































- 


1 


1 


1 


1 


1 


I 


1 


i 


i 


: > 



10 



(b) 



14 



18 



Fig. 10.7.3a, b. Reflection of a bore from a rigid wall 

use of (10.7.4), since h x and u ± = w remain the same on one side of the 
shock, but we must now choose the smallest of the three roots of 



332 



WATER WAVES 



(10.7.3) as the shock speed |_ since the side (1) is now obviously the 
front side of the shock. The depth h 2 of the water behind the shock 
after the reflection — that is, of the water in contact with the wall 
after reflection — is then obtained in the same way as h by using 

(10.7.4) with |_ in place of | + and h 2 in place of h : 

(10.7.4)! (w - |_)^ = - l_h 2 . 

By taking a series of values for w we have determined the ratios 
h 2 /h 1 and h 2 /h as functions of h^h^. That is, the height h 2 of the 
reflected wave has been determined as a function of the ratio of the 
depth h x of the incoming wave to the initial depth h at the wall. The 
results of such a calculation are shown in Figs. 10.7.3a and 10.7.3b: 
In Fig. 10.7.4 we give a curve showing (h 2 — h )/h as a function of 
(h ± — h )/h , that is, we give a curve showing the increase in depth 
after reflection as a function of the relative height (h x — h )/h Q of the 
incoming wave. 



100 
80 


h 2 -h 
h 
































































/ 






60 

40 

20 
















































































1 


1 


1 


1 


1 


1 


1 


1 







10 2 4 6 8 10 12 14 16 

Fig. 10.7.4. Height of the reflected bore 



h,-h c 



For hjho near to unity, i.e. for (h x — h )/h small, it is not difficult 

to show that 

h 2 — h h, — h n 

(10.7.5) — ^2-^ -. 

K h-o 

From this relation we may write h 2 — h c^ 2(h x — h ) if (h^ — h ) is 

small, i.e. the increase in the depth of the water after reflection is 



LONG WAVES IN SHALLOW WATER 



333 



twice the height of the incoming wave when the latter is small. This 
is what one might expect in analogy with the reflection of acoustic 
waves of small amplitude. However, if hJhQ is not small, the water 
increases in depth after reflection by a factor larger than 2. For 
instance, if h^fiQ is 2, then h 2 — h ~ 3^ — h ); while if hJh Q is 10, 
then h 2 — /* — 35(h x — h ), as one sees from the graph of Fig. 
10.7.4. In other words, the reflection of a shock or bore from a rigid 
wall results in a considerable increase in height and hence also in pres- 
sure against the wall if the incoming wave is high. In fact, for very 
high waves the total pressure p per unit width of the wall could be 
shown to vary as the cube of the depth ratio hjh^. 

In the upper curve of Fig. 10.7.3a we have drawn a curve for the 
analogous problem in gas dynamics, i.e. the reflection of a shock from 
the stopped end of a tube. In the case of air with an adiabatic expo- 
nent y = 1.4 the density ratio q 2 /Qi as a function of q 1 /q (in an 
obvious notation) is plotted as a dotted curve in the figure. As we 
see, the density in air on reflection is higher than the corresponding 
quantity, the depth, in the analogous case in water. However, the 
curve for air ends at Qx/qq = ®> since it is not possible to have a shock 
wave in a gas with y = 1.4 which has a higher density ratio. In water 
there is no such restriction. The explanation for this difference lies 
in the fact that the energy law is assumed to hold across a shock in gas 
dynamics, but not in our theory for water waves. 



10.8. The breaking of a dam 

At the end of section 10.5 we gave the solution to an idealized ver- 
sion of the problem of determining the flow which results from the sud- 
den destruction of a dam if it is assumed that the downstream side 




x = 



Fig. 10.8.1. Breaking of a dam 



334 



WATER WAVES 



of the dam is initially free of water. In the present section we consider 
the more general problem which arises when it is assumed that there 
is water of constant depth on the downstream as well as the upstream 
side of the dam. Or, as the situation could also be described: a hori- 
zontal tank of constant cross section extending to infinity in both 
directions has a thin partition at the section x = 0. For x > the 
water has the depth h and for x < the depth h v with h < h ly as 
indicated in Fig. 10.8.1. The water is assumed to be at rest on both 
sides of the dam initially. At the time t = the dam is suddenly des- 
troyed, and our problem is to determine the subsequent motion of 
the water for all x and t. 

The special case h = 0— the cavitation case— was treated, as we 
have already mentioned, at the end of section 10.5. We found there 
that the discontinuity for x = and t = was instantly wiped out 
and that the surface of that portion of the water in motion took the 
form of a parabola tangent to the «-axis (i.e. to the bottom) at the 
point x = — 2 Vghjt = — 2c x t, in which t is the time and c x the wave 




Fig. 10.8.2. Breaking of a dam 



LONG WAVES IN SHALLOW WATER 335 

speed in water of depth h v If h is different from zero we might there- 
fore reasonably expect (on the basis of the discussion at the beginning 
of section 10.6) that a shock wave would develop sooner or later on 
the downstream side, since the water pushing down from above acts 
somewhat like a piston being pushed downstream with an accelera- 
tion. In fact, since the water at x = seems likely to acquire instan- 
taneously a velocity different from zero it is plausible that a shock 
would be created instantly on the downstream side. The simplest 
assumption to make would be that the shock then moves downstream 
w T ith constant velocity | (cf. Fig. 10.8.2). If this were so, the state of 
the water immediately behind the shock (i.e. on the upstream side 
of it) would be constant for all time, since the velocity u 2 and depth 
h 2 on the upstream side of the shock would have the constant values 
determined from the shock relations for the fixed values u = and 
h = h for the velocity and depth on the downstream side and the 
assumed constant value | for the shock velocity. However, it is clear 
that the constant state behind the shock could not extend indefinitely 
upstream since u 2 ^ while the velocity of the water far upstream is 
zero. Since we undoubtedly are dealing with a depression wave be- 
hind the shock it seems plausible to expect that the constant state 
behind the shock changes eventually at some point upstream into a 
centered simple wave of the type discussed in section 10.5. In Fig. 
10.8.2 we indicate in an x, 2-plane a motion which seems plausible as a 
solution of our problem. In the following we shall show that such a 
motion can be determined in a manner compatible with our theory 
for every value of the ratio h^jh^ 

As indicated in Fig. 10.8.2, we consider four different regions in the 
fluid at any time t = t : the zone (0) is the zone of quiet downstream 
which is terminated on the upstream side by the shock wave, or bore; 
the zone (2) is a zone of constant state in which the water, however, is 
not at rest; the zone (3) is a centered simple wave which connects the 
constant state (2) with the constant state (1 ) of the undisturbed water 
upstream. We proceed to show that such a motion exists and to deter- 
mine it explicitly for all values of the ratio h^W between zero and one. 

For this purpose it is convenient to write the shock conditions for 
the passage from the state (0) to the state (2) in the form 

(10.8.1) - l(u 2 - f) = i(4 + 4); 

(10.8.2) c\{u 2 -£) = - cfe 

which are the same conditions as (10.6.18) and (10.6.19) with ggjQ 



336 WATER WAVES 

replaced by c\ = gh { , i.e. by the square of the wave propagation speed 
in water of depth h { . By eliminating c 2 from (10.8.1) by use of (10.8.2) 
and then solving the resulting quadratic for u 2 one readily obtains 

(10.8.3) u 2 /c = i/c - ||l + (VI + 8(£/c )*). 

(The plus sign before the radical was taken in order that u 2 — | and 
— I should have the same sign. We observe also that only positive 
values of | and u 2 are in question throughout our entire discussion 
since the side of (0) is the front side of the shock and the positive 
^-direction is taken to the right.) It is also useful to eliminate u 2 
from (10.8.3) by using (10.8.2); the result is easily put into the form 

(10.8.4) ~={i (Vl +8(£/c )* - l)}i 
c o 

The relations (10.8.3) and (10.8.4) yield the velocity u 2 and the wave 
speed c 2 behind the shock as functions of | and the wave speed c in 
the undisturbed water on the downstream side of the dam. We pro- 
ceed to connect the state (2) by a centered simple wave (cf. the dis- 
cussion in section 10.5) with the state (1). In the present case the 
straight characteristics in the zone (3) (cf. Fig. 10.8.2) are those with 
the slope u — c (rather than the slope u + c as in section 10.5); 
hence the straight characteristics which delimit the zone (3) are the 
lines x = — c x t on the left and x = (u 2 — c 2 )t on the right. Along 
each of the curved characteristics in zone (3) — one of these is indicated 
schematically by a dotted curve in Fig. 10.8.2— the quantity u + 2c 
is a constant; it follows therefore that on the one hand 

(10.8.5) u + 2c = 2c 1 
since u x = 0, while on the other hand 

(10.8.6) u + 2c = u 2 + 2c 2 
throughout the zone (3). The relation 

(10.8.7) u 2 /c + 2c 2 /c = 2c 1 /c 

must therefore hold. Our statement that a motion of the type shown 
in Fig. 10.8.2 exists for every value of the depth ratio hJliQ — or, what 
amounts to the same thing, the ratio c^/cj — is equivalent to the state- 
ment that the relation (10.8.7) furnishes through (10.8.3) and (10.8.4) 
an equation for £/c which has a real positive root for every value of 
cJcq larger than one. This is actually the case. In Fig. 10.8.3 we have 
plotted curves for u 2 /c , 2c 2 /c , and u 2 /c + 2c 2 /c as functions of £/c . 
Once the curves of Fig. 10.8.3 have been obtained, oar problem can 



LONG WAVES IN SHALLOW WATER 337 

be considered solved in principle: From the given value of h^h^ = 
c\\c\ we can determine |/c from the graph (or, by solving (10.8.7)). 
The values of u 2 /c and c 2 /c are then also determined, either from the 
graph or by use of (10.8.3) and (10.8.4). The constant state in the 
zone (2) would therefore be known. In zone (3) the motion can now 
be determined exactly as in section 10.5; we would have along the 
straight characteristics in this zone the relations 

dx x 

— = — = u — c = 2c, — 3c = %u — c x 

at t 

from which 

(10.8.8) c 2 = 1 1 2c x — — ) , and 

(10.8.9) u = l (*i+y). 

Thus the water surface in the zone (3) is curved in the form of a 
parabola in all cases.* At the junctions with both zones (1) and (2) 
the parabola does not have a horizontal tangent, so that the slope of 
the water surface is discontinuous at these points. 

Some interesting conclusions can be drawn from (10.8.8) and 
(10.8.9). By comparison with Fig. 10.8.2 we observe that the Z-axis, 
i.e. the line x = 0, is a characteristic belonging to the zone (3) pro- 
vided that u 2 ^ c 2 since the terminal characteristic of the zone (3) 
on the right lies on the 2-axis or to the right of it in this case. If this 
condition is satisfied we observe from (10.8.8) and (10.8.9)— which 
are then valid on the 2-axis — that c and u are both independent of t 
at x = 0, which means that the depth of the water and its velocity u 
are both independent of / at this point, i.e. at the original location of 
the dam, and hence that the volume of water crossing the original 
dam site per unit of time (and unit of width) dQ/dt = uh is independ- 
ent of time although the motion as a whole is not a steady motion. 
In fact, h = \h Y and u = § c x for all time t at this point. In 
addition, u and c, and thus also dQ/dt, are not only independent of t 
as long as u 2 ^ c 2 , but also independent of the undisturbed depth h 
on the lower side of the dam if h x is held fixed. Of course, it is clear that 
h^\h x must be kept under a certain value (which from section 10.5 
evidently must be less than 4/9) or the condition u 2 ^ c 2 could not be 

* Relations (10.8.8) and (10.8.9) are exactly the same as (10.5.8) and (10.5.7) 
except for a change of sign which arises from a different choice of the positive 
x- direction. 



338 



WATER WAVES 




2 3 4 5 6 

Fig. 10.8.3. Graphical solution for £ 

fulfilled. In fact, the critical value of the ratio h$\h x at which u 2 = c 2 
can be determined easily by equating the right hand sides of (10.8.3) 
and (10.8.4) and determining the value of |/c for this case, after 
which c 2 /c = Vh 2 /h is known from (10.8.4). Since c, 



3 <-i 



in 



the critical case — either from the known fact that we still have 
h 2 = \h x in this case, or from (10.8.8) with x = — we thus are 
able to compute the critical value of c\jcl = h^h^. A numerical cal- 
culation yields for the critical value of the ratio hjh the value 7.225, 
or for hjhj^ the value .1384. Thus if the water depth on the lower side 
of the dam is less than 13.8 percent of the depth above the dam the 
discharge rate on breaking the dam will be independent of the original 
depth on the lower side as well as independent of the time. However, 
if Hq/^ exceeds the critical value .1384, the depth, velocity, and dis- 
charge rate will depend on h ; but they continue to be independent of 



LONG WAVES IN SHALLOW WATER 



339 



the time since the line x = in the as, Z-plane is under the latter cir- 
cumstances contained in the zone (2), which is one of constant state. 

The above results, which at first perhaps seem strange, can be 
made understandable rather easily from the physical point of view, 
as follows. If the zone (3) includes the /-axis (i.e. if hjhj^ is below the 
critical value) we may apply (10.8.8) and (10.8.9) for x = to obtain 
at this point c = u == § c v In other words, the flow velocity at the 
dam site is in this case just equal to the wave propagation speed 
there. For x > 0, i.e. downstream from the dam, we observe from 
(10.8.8) and (10.8.9) that u is greater than c. Since c is the speed at 
which the front of a disturbance propagates relative to the moving 
water we see that changes in conditions below the dam can have no 
effect on the flow above the dam since the flow velocity at all points 
below the dam is greater than the wave propagation speed at these 
points and hence disturbances can not travel upstream. However, 
once h^hi is taken higher than the critical value, the flow velocity 
at the dam will be less than the wave propagation speed at this point, 
as one can readily prove, and we could no longer expect the flow at 
that point to be independent of the initial depth assumed on the 
downstream side. 

The discharge rate dQ/dt = hu per unit width at the dam, i.e. at 
x = 0, is plotted in Fig. 10.8.4 as a function of the depth h . In accord- 
ance with our discussion above we observe that dQ/dt remains con- 



0.3 



0.2 



0.1 



(d-?)/(v 


,) 












- 














- 




\ 










1 


1 


1 


1 


l\ 







0.2 0.4 0.6 0.8 1.0 f*£l 

Fig. 10.8.4. Discharge rate at the dam 



stant at the value dQ/dt = .296/^ until hj^ reaches the critical 
value .138, after which it decreases steadily to the value zero when 
h = h lt i.e. when the initial depth of the water below the dam is the 
same as that above the dam. 



340 



WATER WAVES 



Another feature of interest in the present problem is the height of 
the bore, i.e. the quantity h 2 — h , as a function of the original depth 
ratio h lh v When h = we know that there is no bore and the water 
surface (as we found in section 10.5) appears as in Fig. 10.8.5. The 
water surface at the front of the wave on the downstream side is 
tangent to the bottom and moves with the speed 2c v On the other 
hand, when h^h x approaches the other extreme value, i.e. unity, it is 
clear that the height h 2 — h of the bore must again approach zero. 
Hence the height of the bore must attain a maximum for a certain 




— h, 



r 



Parabola 



x = -c,t x = 2c, t x 

Fig. 10.8.5. Motion down the dry bed of a stream 

value of hjh^ In Fig. 10.8.6 we give the result of our calculations for 
h 2 — h as a function of hjh^ The curve rises very steeply to its 
maximum h 2 — h = .32/^ for h Q lh x = .176 and then falls to zero 
again when h = h v It is rather remarkable that the bore can attain 
a height which is nearly 1/3 as great as the original depth of the water 
behind the dam. 



h 2 -h 
h, 
















0.3 
















/ 














0.2 
0.1 
















/ 




\ 


\ 

























1 


1 


1 


1 


1 \ 




— ►- 



0.2 04 06 0.8 1.0 ho 

h, 

Fig. 10.8.6. Maximum height of the bore 



LONG WAVES IN SHALLOW WATER 



341 



It is instructive to describe the motion by means of the x, Z-plane 
when hjhi is near its two limit values unity and zero. These two cases 
are schematically shown in Fig. 10.8.7. When h ~ h l9 we note that 
the zone (3) is very narrow and that the shock speed | approaches c v 
i.e. the propagation speed of small disturbances in water of depth h lt 
corresponding to the fact that the height of the shock wave tends to 
zero as h ^ h x (cf. Fig. 10.8.6). The other limit situation, i.e. h c± 0, 
is more interesting. Since we tacitly consider h x to remain fixed in our 
present discussion, and hence that h 2 is also fixed since we are in the 
supercritical case, it follows (for example from (10.6.23) with h 2 in 
place of h^) that |->oo as h ^> 0. On the other hand as we see from 




surface for t = t 



(3) 



(2) 



Free surface for t = tQ 



(0) 




Fig. 10.8.7. Limit cases 



Fig. 10.8.6, the height h 2 — h of the shock wave tends to zero rather 
slowly as h -> 0. In the limit, point P becomes the front of the wave 
in accordance with the motion indicated by Fig. 10.8.5. Thus as 
h -> the shock wave becomes very small in height but moves down- 
stream with great speed; or, as we could also say, in the limit the water 
in front of the point P is pinched out and P is the front of the wave. 



342 WATER WAVES 

10.9. The solitary wave 

It has long been a matter of observation that wave forms of a per- 
manent type other than the uniform flows with an undeformed free 
surface occur in nature; for example, Scott Russell [R.14] reported 
in 1844 his observations on what has since been called the solitary 
wave, which is a wave having a symmetrical form with a single hump 
and which propagates at uniform velocity without change of form. 
Later on, Boussinesq [B.16] and Rayleigh [R.3] studied this problem 
mathematically and found approximations for the form and speed of 
such a solitary wave. Korteweg and de Vries [K.15] modified the 
method of Rayleigh in such a way as to obtain waves that are periodic 
in form — called cnoidal waves by them — and which tend to the soli- 
tary wave found by Rayleigh in the limiting case of long wave lengths. 
A systematic procedure for determining the velocity of the solitary 
wave has been developed by Weinstein [W.6]. 

At the beginning of section 10.7 we have shown that the only con- 
tinuous waves furnished by the theory used so far in this chapter 
which progress unchanged in form are of a very special and rather 
uninteresting character, i.e., they are the motions with uniform velo- 
city and horizontal free surface.* This would seem to be in crass con- 
tradiction with our intention to discuss the solitary wave in terms of 
the shallow water theory, and it has been regarded by some writers as a 
paradox. **The author's view is that this paradox— like most others- 
becomes not at all paradoxical when properly examined. What is 
involved is a matter of the range of accuracy of a given approximate 
theory, and also the fact that a perturbation or iteration scheme of 
universal applicability does not exist: one must always modify such 
schemes in accordance with the character of the problem. In the pre- 
sent case, the salient fact is that the theory used so far in the present 
chapter represents the result of taking only the lowest order terms in 
the shallow water theory as developed in section 4 of Chapter 2, and 
it is necessary to carry out the theory to include terms of higher order 

* If motions with a discontinuity are included in the discussion, then the motion 
of a bore is the only other possibility up to now in this chapter with regard to 
waves propagating unchanged in form. 

** Birkhoff [B.ll, p. 23], is concerned more about the fact that the shallow 
water theory predicts that all disturbances eventually lead to a wave which 
breaks when on the other hand Struik [S.29] has proved that periodic progressing 
waves of finite amplitude exist in shallow water. In the next section the problem 
of the breaking of waves is discussed. Ursell [U.ll] casts doubt on the validity 
of the shallow water theory in general because it supposedly does not give rise 
to the solitary wave. 



LONG WAVES IN SHALLOW WATER 343 

if one wishes to obtain an approximation to the solution of the problem 
of the solitary wave. This has been done by Keller [K.6], who finds 
that the theory of Friedrichs [F.ll] presented in Chapter 2, when car- 
ried out to second order,* yields both the solitary wave and cnoidal 
waves of the type found by Korteweg and de Vries [K.15] (thus the 
shallow water theory is capable of yielding periodic progressing waves 
of finite amplitude). As lowest order approximation to the solution 
of the problem, Keller finds (as he must in view of the remarks above), 
that the only possibility is the uniform flow with undeformed free 
surface, but if the speed U of the flow is taken at the critical value 
U = Vgh with h the undisturbed depth, then a bifurcation phenome- 
non occurs (that is, among the set of uniform flows of all depths and 
velocities, the solitary wave occurs as a bifurcation from the special 
flow with the critical velocity) and the second order terms in the de- 
velopment of Friedrichs lead to solitary and cnoidal waves with 
speeds in the neighborhood of this value. To clinch the matter, it has 
been found by Friedrichs and Hyers [F.13] that the existence of the 
solitary wave can be proved rigorously by a scheme which starts with 
the solution of Keller as the term of lowest order and proceeds by 
iterations with respect to a parameter in essentially the same manner 
as in the general shallow water theory.** In the following, we shall 
derive the approximation to the solution of the solitary wave problem 
following the method of Friedrichs and Hyers rather than the general 
expansion scheme which was used by Keller, and we can then state 
the connection between the two in more detail. 

The author thus regards the nonlinear shallow water theory to be 
well founded and not at all paradoxical. Indeed, the linear theory of 
waves of small amplitude treated at such length in Part II of this 
book is in essentially the same position as regards rigorous justifica- 
tion as is the shallow water theory: we have only one or two cases so 
far in which the linear theory of waves of small amplitude is shown to 
be the lowest order term in a convergent development with respect to 
amplitude. We refer, in particular, to the theory of Levi-Civita [L.7] 
and Struik [S.29] in which the former shows the existence of periodic 
progressing waves in water of infinite depth and the latter the same 
thing (and by the same method) for waves in water of finite constant 

* In order to fix all terms of second order, Keller found it necessary to make 
use of certain relations which result from carrying the development of some of 
the equations up to terms of third order. 

** W. Littman, in a thesis to appear in Communs. Pure and Appl. Math., 
has proved rigorously in the same way the existence also of cnoidal waves. 



344 WATER WAVES 

depth.* This theory will be developed in detail in Chapter 12. It might 
be added that those who find the nonlinear shallow water theory 
paradoxical in relation to the solitary wave phenomenon should by 
the same type of reasoning also find the linear theory paradoxical, 
since it too fails to yield any approximation to the solitary wave, even 
when carried out to terms of arbitrarily high order in the amplitude, 
except the uniform flow with undisturbed free surface. In fact, if one 
were to assume that a development exists for the solitary wave which 
proceeds in powers of the amplitude as in the theory discussed in the 
first part of Chapter 2, it is easily proved that the terms of all orders 
in the amplitude are identically zero. There is no paradox here, how- 
ever; rather, the problem of the solitary wave is one in which the 
solution is not analytic in the amplitude in the neighborhood of its 
zero value, but rather has a singularity— possibly of the type of a 
branch point — there. Thus a different kind of development is needed, 
and, as we have seen, one such possibility is a development of the type 
of the shallow water theory starting with a nonlinear approximation. 
Another possibility has been exploited by Lavrentieff [L.4] in a 
difficult paper; Lavrentieff proves the existence of the solitary wave 
by starting from the solutions of the type found by Struik for periodic 
waves of finite amplitude and then making a passage to the limit by 
allowing the wave length to become large and, presumably, in such a 
way that the parameter gh/U 2 tends to unity. This procedure of 
Lavrentieff thus also starts with a nonlinear first approximation. 
The problem thus furnishes another good example of the well-known 
fact that it is not always easy to guess how to set up an approximation 
scheme for solving nonlinear boundary value problems, since the 
solution may behave in quite unexpected ways for particular values of 
the parameters. Hindsight, however, can help to make the necessity 
for procedures like those of Friedrichs and Hyers and of Lavrentieff 
in the present case more apparent: we have seen in Chapter 7.4 that 
a steady flow with the critical speed U = vgh is in a certain sense 
highly unstable since the slightest disturbance would lead, in terms 
of the linear theory for waves of small amplitude, to a motion in which 
infinite elevations of the free surface would occur everywhere; thus 
the linear theory of waves of small amplitude seems quite inappro- 
priate as the starting point for a development which begins with a 



* L. Nirenberg [N.2] has recently proved the existence of steady waves of 
finite amplitude caused by flows over obstacles in the bed of a stream. 



LONG WAVES IN SHALLOW WATER 



345 



uniform flow at the critical speed, and one should consequently use 
a basically nonlinear treatment from the outset. 

We turn now to the discussion of the solution of the solitary wave 
problem. The theory of Friedrichs and Hyers begins with a formula- 
tion of the general problem that is the same as that devised by Levi- 
Civita for treating the problem of existence of periodic waves of finite 
amplitude, and which was motivated by the desire to reformulate the 
problem in terms of the velocity potential op and stream function \p as 
independent variables in order to work in the fixed domain between 
the two stream lines xp = const, corresponding to the bottom and the 
free surface instead of in the partially unknown domain in the physi- 
cal plane. We therefore begin with the general theory of irrotational 
waves in water when a free surface exists. The wave is assumed to 
be observed from a coordinate system which moves with the same 
velocity as the wave, and hence the flow can be regarded as a steady 
flow in this coordinate system. 




Fig. 10.9.1. The solitary wave 
A complex velocity potential %'(x',y') = %'(z'): 

(10.9.1) *' = ?>'+%'> z' = x'+iy f 

is sought in an x\ ?/'-plane (cf. Fig. 10.9.1) such that at infinity the 
velocity is U and the depth of the water is h. %' is of course an analytic 
function of z' . The real harmonic functions cp' and ip f represent the 
velocity potential and the stream function. The complex velocity 
w' = d%'/dz' is given by 

(10.9.2) w' = u' — iv', 

in which u' and v' are the velocity components. This follows by virtue 
of the Cauchy-Riemann equations: 

(10.9.3) 9>V=VV> PV = -VV. 



346 WATER WAVES 

since w' = cp' x > -f- *V*' # ^ * s convenient to introduce new dimensionless 
variables: 

(10.9.4) z = z'/h, w = K>'/*7, £ = cp + iy> = x'K hU )> 
and a parameter y: 

(10.9.5) y = gh/U 2 . 

In terms of these quantities the free surface corresponds to \p — 1 
if the bottom is assumed given by ip = 0, since the total flow over a 
curve extending from the bottom to the free surface is Uh. The 
boundary conditions are now formulated as follows: 

(10.9.6) v = — Jmw = at ip = 0, 

(10.9.7) J \w\ 2 +yy = const. at y; = 1. 

The second condition results from Bernoulli's law on taking the 
pressure to be constant at the free surface and the density to be unity, 
as one sees from equation (1.3.4) of Ch. 1. At oo we have the condition 

(10.9.8) w -> 1 as | x | -> oo. 

We assume now that the physical plane (i.e. the x, i/-plane) is mapped 
by means of ^(2) into the 99, y;-plane in such a way that the entire flow 
is mapped in a one-to-one way on the strip bounded by ip = and 
y) = 1.* In this case the inverse mapping function z(%) exists, and we 
could regard the complex velocity w as a function of # defined in the 
strip bounded by xp = 0, \p = 1 in the #-plane. We then determine the 
analytic function w(%) in that strip from the boundary conditions 
(10.9.6), (10.9.7), (10.9.8), after which %(z) can be found by an inte- 
gration and the free surface results as the curve given by y) = *fm #=1. 
It is convenient, however, again following Levi-Civita to replace 
the dependent variable w by another (essentially its logarithm) 
through the equation 

(10.9.9) w = e - i{Q+a) . 
It follows that 

(10.9.10) |w| = e\ 6 = argw, 

and thus X = log |re>|, with \w\ the magnitude of the velocity vector, 

* Our assumption that the mapping of the flow on the ^-plane is one-to-one 
can be shown rather easily to follow from the other assumptions and Levi-Civita 
carries it out. The equivalence of the various formulations of the problem is then 
readily seen. In Chapter 12.2 these facts are proved. 



LONG WAVES IN SHALLOW WATER 347 

and d is the inclination relative to the #-axis of the velocity vector. 
We proceed to formulate the conditions for the determination of 
and X in the <p, rp-plane. The condition (10.9.6) becomes, of course, 
$ = at ip = 0. To transform the condition (10.9.7) we first differen- 
tiate with respect to <p along the line y) = 1 to obtain 

d | w I dy 

(10.9.11) | w | — y^ + Y Y = ° on y> = 1. 

Since x and y are conjugate harmonic functions of cp and \p we may 
write 

dx dy (p x u 

12 



(10.9.12) 



d<p dip <pl + <p 2 y \w T 
dy dx v 



dcp dip | w | 2 

in accordance with well-known rules for calculating the derivatives 
of functions determined implicitly, or from 

dz 1 1 u + iv 

dx dx u — iv u 2 -\- v 2 ' 
dz 
As a consequence we have from (10.9.11): 

d\w\ 

or, since | w \ = e x and v = — <fm e~ i{6+a) — e K sin 6: 

7)} 
e x ^— = — ye~ 2k sin 6, 
dcp 

and since dX\d<p = —dd/dip because X and 6 are harmonic conjugates 
it follows finally that 

dd 
(10.9.13) — = ye- u sind attp = 1. 

The boundary conditions 6 = for tp = and (10.9.13) at \p = 1 are 
Levi-Civita's conditions, but the condition at oo imposed here is re- 
placed in Levi-Civita's and Struik's work by a periodicity condition in 
x, — and this makes a great difference. Levi-Civita and Struik proceed 
on the assumption that a disturbance of small amplitude is created 
relative to the uniform flow in which w = const.; this is interpreted to 



348 WATER WAVES 

mean that 6 + iX is a quantity which can be developed in powers of 
a small parameter e, and the convergence of the series for sufficiently 
small values of e is then proved. In Chapter 12.2 we shall give a proof 
of the convergence of this expansion. (In lowest order, we note that 
the condition (10.9.13) leads for small X and 6 to the condition dO/dip — 
yO = at xp — 1— in agreement with what we have seen in Part II.) 
In the case of the solitary wave such a procedure will not succeed, as 
was explained above, or rather it would not yield anything but a 
uniform flow. The procedure to be adopted here consists in developing, 
roughly speaking, with respect to the parameter y near y = 1; but, 
as in the shallow water theory in general in the version presented in 
section 4 of Chapter 2, we introduce a stretching of the horizontal 
coordinate <p which depends on y while leaving the vertical coordinate 
unaltered (see equation (10.9.19)). This stretching of only one of the 
coordinates is the characteristic feature of the shallow water theory. 
(The approximating functions are then no longer harmonic in the new 
independent variables.) Specifically, we introduce the real parameter 
x by means of the equation 

(10.9.14) e- 3 * 2 = y = gh/U 2 . 

This implies that gh/U 2 < 1, but that seems reasonable since all of 
the approximate theories for the solitary wave lead to such an 
inequality. We also introduce a new function t, replacing X, by the 
relation 

(10.9.15) t = X+x 2 . 

For 0((p t ip) and r((p, ip) we then have the boundary conditions 

(10.9.16) 6 = 0, tp = 

(10.9.17) — = 6T 3T sin0, w = l. 
dtp 

For <p -> i oo we have the conditions imposed by the physical pro- 
blem: 

(10.9.18) 6 -+0, t ^x 2 , 

the latter resulting since X -> at oo from \ w \ = e x and | w \ -> 1 
at oo. As we have already indicated, the development we use requires 
stretching the variable op so that it grows large relative to y) when x 
is small; this is done in the present case by introducing the new in- 
dependent variables 



LONG WAVES IN SHALLOW WATER 349 

(10.9.19) <p = x<p, ip = ip. 

The dependent variables 6 and t are now regarded as functions of <p 
and ip and they are then expanded in powers of x: 

x = 9c 2 r x (% ip) + x i r 2 (<p, $)+..., 



(10.9.20) ( Q = ^ 6 ^ _ } + ^. Q ^ - } + ._ 

(We have omitted writing down a number of terms which in the 
course of the calculation would turn out to have zero coefficients.) 
Friedrichs and Hyers have proved that the lowest order terms in 
these series, as obtained formally through the use of the boundary 
conditions, are the lowest order terms in a convergent iteration scheme 
using x as small parameter. Their convergence proof also involves the 
explicit use of the stretching process. However, the proof of this 
theorem is quite complicated, and consequently we content ourselves 
here with the determination of the lowest order terms: we remark, 
however, that higher order terms could also be obtained explicitly 
from the formal expansion. 

The series in (10.9.20) are now inserted in all of the equations which 
serve to determine 6 and t, and relations for the coefficient functions 
T t -(9?, ip) and Oi((p, ip) are obtained. The Cauchy-Riemann equations 
for Q and r lead to the equations 

(10.9.21) 0_ = _^ T _ j r- w = xd- 

in terms of the variables 99 and ip, and the series (10.9.20) then yield 
the equations 

(10.9.22) Tl - = 0, 6 { - = - r,- , t 2 - = 0& . 

Thus r x = Ti(^) is independent of ip, and integration of the remaining 
equations gives the following results: 

fd 1 = — tpx[ 
(10-9.23) r 2 = -Wt'; +j(<p) 

[e 2 =■ !^ 3 t;" - ipf(f). 

The primes refer to differentiation with respect to (p. An additive 
arbitrary function of q> in the first of these equations was taken to be 
zero because of the boundary condition d x = for ip = 0. 

Upon substitution of (10.9.20) into the boundary condition (10.9. 
17) we find 

x 3 0^ + x 5 d 2 - + . . . = x 3 d 1 - S* 5 ^! + x 5 Q 2 + . . . 



350 WATER WAVES 

and consequently we have the equations 

(10.9.24) aty) = l. 

I #2y = 2 — ^r 1 d 1 , 

The first equation is automatically satisfied because of the first equa- 
tion of (10.9.23). The second equation leads through (10.9.23) to the 
condition 

(10.9.25) tJ" = 9-^Ti, 

for r v as one readily verifies. Once r x has been determined, one sees 
that d x is also immediately fixed by the first equation in (10.9.23). 
Boundary conditions are needed for the third order nonlinear differ- 
ential equation given by (10.9.25); we assume these conditions to be 

(Ti'(0) =0, 

(10.9.26) I t x (oo) = 1, 

Ui'(oo) = 0. 

These conditions result from our assumed physical situation: the first 
is taken since a symmetrical form of the wave about its crest is ex- 
pected and hence 6^0) = 0, the second arises from (10.9.18), while 
the third is a reasonable condition that is taken in place of what looks 
like the more natural condition Tj(oo) = since the latter condition 
is automatically satisfied, in view of the first equation of (10.9.23) 
and 0x(oo) = 0, and thus does not help in fixing r 1 uniquely. 
An integral of (10.9.25) is readily found; it is: 

r'l = |tJ + const., 

and the boundary conditions yield 

ri'=!(T?-D. 
From this one obtains, finally, the solution: 

(10.9.27) Tj(^) = 1-3 sech 2 (3<p/2), 

and ± is then fixed by (10.9.23). From these one finds for the shape of 
the wave— that is, the value of y corresponding to y> = 1 — , and for 
the horizontal component u of the velocity the equations 

(10.9.28) y = 1 +3^ 2 sech 2 , 

A 

3xx 

(10.9.29) u == 1 — 3* 2 sech 2 . 



LONG WAVES IN SHALLOW WATER 



351 




Fig. 10.9.2. A solitary wave 



In calculating these quantities, 
higher order terms in x have been 
neglected. The expression for the 
wave profile is identical with those 
found by Boussinesq, Rayleigh, 
and Keller. For the velocity u, 
the two former authors give u = 1 
while Keller gives the same expres- 
sion as above except that the factor 
3x 2 is replaced by another which 
differs from it by terms of order 
^ 4 or higher. 

Thus a solitary wave of sym- 
metrical form has been found with 
an amplitude which increases with 
its speed U. Careful experiments 
to determine the wave profile and 
speed of the solitary wave have 
been carried out by Daily and 
Stephan [D.I], who find the wave 
profile and velocity to be closely 
approximated by the above formu- 
las with a maximum error in the 
latter of 2.5 % at the highest ampli- 
tude-depth ratio tested. Fig. 10.9.2 
is a picture of a solitary wave taken 
by Daily and Stephan; three frames 
from a motion picture film are 
shown. 



10.10. The breaking of waves in shallow water. Development of bores 

In sections 10.4 and 10.6 above it has already been seen that the 
shallow water theory, which is mathematically analogous to the 
theory of compressible flows in a gas, leads to a highly interesting 
and significant result in cases involving the propagation of disturb- 
ances into still water that are the exact counterparts of the corre- 
sponding cases in gas dynamics involving the motions due to the action 
of a piston in a tube filled with gas. These cases, which are very easily 



352 WATER WAVES 

described in terms of the concept of a simple wave (cf. sec. 10.3), 
lead, in fact to the following qualitative results (cf. sec. 10.4): there is 
a great difference in the mode of propagation of a depression wave 
and of a hump with an elevation above the undisturbed water line; 
in the first case the depression wave gradually smooths out, but in 
the second case the front of the wave becomes progressively steeper 
until finally its slope becomes infinite. In the latter case, the mathe- 
matical theory ceases to be valid for times larger than those at which 
the discontinuity first appears, but one expects in such a case that the 
wave will continue to steepen in front and will eventually break. This 
is the correct qualitative explanation, from the point of view of 
hydrodynamical theory, for the breaking of waves on shallow beaches. 
It was advanced by Jeffreys in an appendix to a book by Cornish 
[C.7] published in 1934. Jeffreys based his discussion on the fact that 
the propagation speed of a wave increases with increase in the height 
of a wave above the undisturbed level. Consequently, if a wave is 
created in such a way as to cause a rise in the water surface it follows 
that the higher points on the wave surface will propagate at higher 
speed than the lower points in front of them— in other words there is 
a tendency for the higher portions of the wave to overtake and to 
crowd the lower portions in front so that the front of the wave be- 
comes steep and eventually curls over and breaks; the same argument 
indicates that a depression wave tends to flatten out and become 
smoother as it advances. 

It is of interest to recall how waves break on a shallow beach. 
Figures 10.10.1, 10.10.2, and 10.10.3 are photographs* of waves on 
the California coast. Figure 10.10.1 is a photograph from the air, 
taken by the Bureau of Aeronautics of the U.S. Navy, which shows 
how the waves coming from deep water are modified as they move 
toward shore. The waves are so smooth some distance off shore that 
they can be seen only vaguely in the photograph, but as they move 
inshore the front of the waves steepens noticeably until, finally, 
breaking occurs. Figures 10.10.2 and 10.10.3 are pictures of the same 
wave, with the picture of Figure 10.10.3 taken at a slightly later time 
than the previous picture. The steepening and curling over of the 
wave are very strikingly shown. 

At this point it is useful to refer back to the beginning of section 
10.6 and especially to Fig. 10.6.1. This figure, which is repeated here 

* These photographs were very kindly given to the author by Dr. Walter Munk 
of the Scripps Institution of Oceanography. 



LONG WAVES IN SHALLOW WATER 



353 




Fig. 10.10.1. Waves on a beach 



for the sake of convenience, indicates in terms of the theory of 
characteristics what happens when a wave of elevation is created by 




Fig. 10.10.2. Wave beginning to break 



354 



WATER WAVES 







Fig. 10.10.3. Wave breaking 

pushing the moveable end of a tank of water into it so that a disturb- 
ance propagates into still water of constant depth: the straight 
characteristics issuing from the "piston curve" AD, along each of 




Fig. 10.10.4. Initial point of breaking 



LONG WAVES IN SHALLOW WATER 355 



which the velocity u and the quantity c = Vg(h -f v) are constant, 
eventually intersect at the point E. The point E is a cusp on the enve- 
lope of the characteristics, and represents also the point at which the 
slope of the wave surface first becomes infinite. The point E might 
thus— somewhat arbitrarily, it is true— be taken as defining the break- 
ing point (x b , t b ) of the wave, since one expects the wave to start 
curling over after this point is reached. It is possible to fix the values 
of x b and t b without difficulty once the surface elevation r\ — 7/(0, t) is 
prescribed at x = 0; we carry out the calculation for the interesting 
case of a pulse in the form of a sine wave: 

(10.10.1) f)(0, t) = A sin co*. 

For t = 0, x > we assume the elevation r\ of the water to be zero 
and its velocity u to be constant (though not necessarily zero, since 
it is of interest to consider the effect of a current on the time and place 
of breaking). 

As we know, the resulting motion is easily described in terms of the 
characteristics in the x, /-plane, which are straight lines emanating 
from the /-axis, as indicated in Figure 10.10.6. The values of u and c 
are constant along each such straight line. The slope dx/dt of any 
straight characteristic through the point (0, t) is given by 

dx 

(10.10.2) — = 3c - 2c + u Q , 

which is the same as (10.3.6). The quantity c has the value c = Vgh, 
while c = Vg(h -{- rj), as always. On the other hand, the slope of this 
characteristic is clearly also given in terms of a point (x, t) on it by 
x/(t — t) so that (10.10.2) can be written in the form 

(10.10.3) x = {t — t)[3c(t) — 2c -f u ] 

in which we have indicated explicitly that c depends only on t since 
it (as well as all other quantities) is constant along any straight 
characteristic. Thus (10.10.3) furnishes the solution of our problem, 
once c(t) is given, throughout a region of the x, /-plane which is cover- 
ed by the straight lines (10.10.3) without overlapping. However, the 
interesting cases for us are just those in which overlapping occurs, 
i.e. those for which the characteristics converge and eventually cut 
each other, and this always happens if an elevation is created at 
x = 0. In fact, if c is an increasing function of t, then dx/dt as given 
by (10.10.2) increases with t and hence the characteristics for x > 



356 WATER WAVES 

must intersect. In this case, furthermore, the family of straight cha- 
racteristics has an envelope beginning at a point (cc b , t b ), which we 
have defined to be the point of breaking. 

We proceed to determine the envelope of the straight lines (10.10.3). 
As is well known, the envelope can be obtained as the locus resulting 
from (10.10.3) and the relation 

(10.10.4) = - [3c(t) - 2c + u ] + 8(* - t)c'(t) 

obtained from it by differentiation with respect to r. For the points 
(x c , t c ) on the envelope we then obtain the parametric equations 

(10.10.5) * c = -— , 

and 

[3c(t) — 2c + u ] 

(lo.io.e, te = r + LAl_JL±_°i. 

We are interested mainly in the "first" point on the envelope, that 
is, the point (x b , t b ) for which t c has its smallest value since we iden- 
tify this point as the point of breaking. To do so really requires a 
proof that the water surface has infinite slope at this point. Such a 
proof could be easily given, but we omit it here with the observation 
that an infinite slope is to be expected since the characteristics which 
intersect in the neighborhood of the first point on the envelope all 
carry different values for c. 

We have assumed that r](0, t) is given by (10.10.1 ) and consequently 
the quantity c(r) in (10.10.5) and (10.10.6) is given by 



(10.10.7) c(t) = Vg(h + A sin cot). 

If we assume A > we see that c'(r) is a positive decreasing function 
of r for small positive values of t. Since c(r) increases for small posi- 
tive values of r it follows that both x c and t c in (10.10.5) and (10.10.6) 
are increasing functions of t near t = 0. A minimum value of x c and 
t c must therefore occur for t — 0, so that the breaking point is given 

by 

2c ( c o + M o) 2 

< io - io - 8) *> = : gA(a • 

and 

2c (c + u ) 



(10.10.9) t b 

v } b 3gAco 

as one can readily verify. We note that the point (x b , t b ) lies on the 



LONG WAVES IN SHALLOW WATER 357 

initial characteristic x = (c + u )t, as it should since r = for this 
characteristic. From the formulas we can draw a number of interesting 
conclusions. Since c = Vgh we see that breaking occurs earlier in 
shallower water for a pulse of given amplitude A and frequency co. 
Breaking also occurs earlier when the amplitude and frequency are 
larger. It follows that short waves will break sooner than long waves, 
since longer waves are correlated with lower frequencies. Finally we 
notice that early breaking of a wave is favored by small values for 
u , the initial uniform velocity of the quiet water. In fact, if u is 
negative, i.e. if the water is flowing initially toward the point where 
the pulse originates, the breaking can be made to occur more quickly. 
Everyone has observed this phenomenon at the beach, where the break- 
ing of an incoming wave is often observed to be hastened by water 
rushing down the beach from the breaking of a preceding wave. 

It is of some importance to draw another conclusion from our theory 
for waves moving into water of constant depth: an inescapable con- 
sequence of our theory is that the maxima and minima of the surface 
elevation propagate into quiet water unchanged in magnitude with re- 
spect to both distance and time. This follows immediately from the fact 
that the values of the surface elevation are constant along the straight 
characteristics so that if r\ has a relative maximum for x = 0, t = r, 
say, then this value of r\ will be a relative maximum all along the 
characteristic which issues from x = 0, t = r. The waves change 
their form and break, but they do so without changes in amplitude. 

In a report of the Hydrographic Office by Sverdrup and Munk 
[S.36] some results of observations of breakers on sloping beaches are 
given in the form of graphs showing the ratio of breaker height to 
deep water amplitude and the ratio of undisturbed depth at the break- 
ing point to the deep water amplitude as functions of the "initial 
steepness" in deep water, the latter being defined as the ratio of 
amplitude to wave length in deep water. The "initial steepness" is thus 
essentially the quantity Aco in our above discussion, and our results 
indicate that it is a reasonable parameter to choose for discussion of 
breaking phenomena. The graphs given in the report — reproduced here 
in Figures 10.10.5a and 10.10.5b — show very considerable scattering 
of the observational data, and this is attributed in the report to errors 
in the observations, which are apparently difficult to make with 
accuracy. On the basis of our above conclusion — that the breaking of 
a wave in water of uniform depth occurs no matter what the amplitude 
of the wave may be in relation to the undisturbed depth— we could 



358 



WATER WAVES 



offer another explanation for the scatter of the points in Figures 
10.10.5a and 10.10.5b, i.e. that the amplitude ratios are relatively 
independent of the initial steepness. Of course, the curves of Figures 



3-0 



2.5 



2.0 



1.5 



1.0 



0.5 













1 i 

SLOPE 
O S.I.O. 


















• W.H.0.1. 








• 










A B.E.B — I: 6.3 








.• 










A B.E.B — 1:20.4 








- 


o 
o 
o 


o • 

> « 

o 


o 

oo 

8* 


o 

°o°°° 


THEORY 




- — <» 

















X°aV 

i 


1 

1 









0.003 0.005 0.01 



0.02 0.03 0.05 



0.1 0.15 



Fig. 10.10.5a. Ratio of breaker height to wave height in deep water, H b /H ', 
assuming no refraction 



4.0 



3.5 



3.0 



X 2.5 



2.0 



1.5 



1.0 



,^\ 


/H^ = 6.66 

• 


• 




— ■ 1 

O S.I.O. 
• W.H.O 
A B.E.B 
A B.E.B 


SLOPE 

— 1 : 6.3 

— 1:20.4 






• 




• • 

•o 


c 


C 



•. 

& 

oo 

o 


i B.E.B 


— 1 : 33.2 








- 


o° 
o 

1 


o 


o 66x, 

8>? 
%° 

1 1 


o '\o 

o^cg 5 


On O 

% ° 

X^a a 

A "~ 
O 

* A* 


a 

AA_^A 
"a A 

1 C 


A 
M A 


l i 


/ ->' 
/ <■ 
/ o| 

Ol _J 
Ull 

I 


0.0 


03 0.0 


05 


0.( 


31 0.0 


2 


.03 


.05 





1 0.15 



Fig. 10.10.5b. Ratio of depth of water at point of breaking to wave height in 
deep water, d b /H ', as function of steepness in deep water, H 'IL , assuming 

no refraction 



LONG WAVES IN SHALLOW WATER 



359 



10.10.5a and 10.10.5b refer to sloping beaches and hence to cases in 
which the wave amplitudes increase as the wave moves toward shore; 
but still it would seem rather likely that the amplitude ratios would be 
relatively independent of the initial steepness in these cases also since 
the beach slopes are small. The detailed investigation of breaking of 
waves by Hamada [H.2], which is both theoretical and experimental 
in character, should be consulted for still further analysis of this and 
other related questions. The papers by Iversen [1.6] and Suquet 
[S.31] also give experimental results concerning the breaking of waves. 
We continue by giving the results of numerical computations for 
three cases of propagation of sine pulses into still water of constant 
depth. The cases calculated are indicated in the following table: 



Case 


Type of pulse 


1 


^^ 




2 


^_^ 


^ 


3 


x— x /-— x x— \ 


" " " 



Case 1 is a half-sine pulse in the form of a positive elevation, case 2 
is a full sine wave which starts with a depression phase, and case 3 
consists of several full sine waves. 

Figure 10.10.6 shows the straight characteristics in the x, /-plane 
for case 1. (In all of these cases, the quantities x and y are now certain 
dimensionless quantities, the definitions of which are given in [S.19].) 
We observe that the envelope begins on the initial characteristic in 
this case, in accord with earlier developments in this section. The 
envelope has two distinct branches which meet in a cusp at the 
breaking point (x b , t h ). Figure 10.10.7 gives the shape of the wave for 
two different times. As we see, the front of the wave steepens until it 
finally becomes vertical for x = x b , t = t b , while the back of the wave 
flattens out. The solution given by the characteristics in Figure 



360 



WATER WAVES 




Fig. 10.10.6. Characteristic diagram in the x, Z-plane 




Fig. 10.10.7. Wave height versus distance for a half-sine wave of amplitude 
A /5 in water of constant depth at two instants, where h n is the height of the 

still water level 



LONG WAVES IX SHALLOW WATER 



361 



•n 



Fig. 10.10.8. Wave profile after breaking 




Fig. 10.10.9. Characteristic diagram in the x, i-plane 



362 



WATER WAVES 



10.10.6 is not valid for x > x b , t > t bi and we expect breaking to 
ensue. However, we observe that the region between the two branches 
of the envelope is quite narrow, so that the influence of the developing 





Fig. 10.10.10. Wave height versus distance for a full negative sine wave with 
amplitude h Q /5 in water of constant depth at t = 3.0, t = 5.0, and t — 6.28 

breaker may not seriously affect the motion of the water behind it. 
Thus we might feel justified in considering the solution by characteris- 
tics given by Figure 10.10.6 as being approximately valid for values of 



LONG WAVES IN SHALLOW WATDR 



363 



t slightly greater than t b . (This also seems to the writer to be intuitive- 
ly rather plausible from the mechanical point of view.) Figure 10.10.8 
was drawn on this basis for a time considerably greater than t b . The 
full portion of the curve was obtained from the characteristics outside 
the region between the branches of the envelope, while the dotted 
portion— which is of doubtful validity— was obtained by using the 
characteristics between the branches of the envelope in an obvious 
manner. In this way one is able to approximate the early stages of the 
curling over of a wave. 

Figures 10.10.9, 10.10.10, and 10.10.11 refer to case 2, in which a 
depression phase precedes a positive elevation. In this case the enve- 
lope of the characteristics does not begin, of course, on the initial 
characteristic but rather in the interior of the simple wave region, as 
indicated in Figure 10.10.9. Figure 10.10.10 shows three stages in the 
progress of the pulse into still water. The steepening of the wave front 
is very marked by the time the breaking point is reached— much more 
marked than in the preceding case for which no depression phase oc- 
curs in front. Figure 10.10.11 shows the shape of the wave a short time 
after passing the braking point. This curve was obtained, as in the 
preceding case, by using the characteristics between the branches of 
the envelope. Although this can yield only a rough approximation, still 




Fig. 10.10.11. r] versus x at t = 7 for non-sloping bottom where the pulse is an 

entire negative sine- wave. The dotted part of the curve represents r\ in the region 

between the branches of the envelope 



one is rather convinced that the wave really would break very soon 
after the point we have somewhat arbitrarily defined as the breaking 
point. 



364 WATER WAVES 

Figure 10.10.12 shows the water surface in case 3 for a time well 
beyond the breaking point. 




Fig. 10.10.12. Water profile after breaking 

In gas dynamics where u and c represent the velocity and sound 
speed throughout an entire cross section of a tube containing the gas, 
it clearly is not possible to give a physical interpretation to the region 
between the two branches of the envelope in the cases analogous to 
that shown in Figure 10.10.6, since the velocity ana propagation 
speed must of necessity be single-valued functions of x. However, in our 
case of water waves u and c refer to values on the water surface so that 
there is no reason a priori to reject solutions for u and c which are not 
single-valued in x. Thus we might be tempted to think that the dotted 
part of the curve in Figure 10.10.8 is valid within the general frame- 
work of our theory, but this is, unfortunately, not the case: our fun- 
damental differential equations are not valid in the "overhanging" 
part of the wave, simply because that part is not resting on a rigid 
bottom. It may be that one could pursue the solutions beyond the 
point where the breaking begins by using the appropriate differential 
equations in the overhanging part of the wave and then piecing to- 
gether solutions of the two sets of differential equations so that con- 
tinuity is preserved, but this would be a problem of considerable 
difficulty. In this connection, however, it is of interest to report the 
results of a calculation by Biesel [B.10] for the change of form of 
progressing waves over a beach of small slope. Not the least interesting 
aspect of Biesel's results is the fact that they are based essentially on 
the theory of waves of small amplitude, i.e. on the type of theory 
which forms the basis for the discussions in Part II of this book. 
However, in Part II only the so-called Eulerian representation was 
used, in which the dependent quantities such as velocity, pressure, 



LONG WAVES IN SHALLOW WATER 365 

etc., are all obtained at fixed points in space. As a result, when lineari- 
zations are introduced the free surface elevation rj, for example, is 
a function of x and t and must of necessity be single-valued. Biesel, 
however, observes that one can also use the Lagrangian representa- 
tion* just about as conveniently as the Eulerian when a development 
with respect to amplitude is contemplated. In this approach, all quan- 
tities are fixed in terms of the initial positions of the water particles 
(and the time, of course). In particular, the displacements (£, rj) of 
the water particles on the free surface would be given as functions of 
a parameter, i.e. £ = £(a, t), rj =rj(a, t), and there would be no necessity 
a priori to require that rj should be a single-valued function of x. 
Biesel has carried out this program with the results shown in Figs. 
10.10.13 to 10.10.16 inclusive. A sinusoidal progressing wave in 
deep water is assumed. The first two figures refer to the theory when 
carried out only to first order terms in the displacements relative to the 
rest position of equilibrium. The second figure is a detail of the motion 
in a neighborhood of the location shown by the dotted circle in the 
first figure. Fig. 10.10.15 and Fig. 10.10.16 treat the same problem, 
but the solution is carried to second order terms. In both cases the 
development of a breaker is strikingly shown. A comparison of the 
results of the first order and second order theories is of interest; the 
main conclusions are: if second order corrections are made the break- 
ing is seen to occur earlier (i.e. in deeper water), the height of the wave 
at breaking is much greater, and the tendency of the wave to plunge 
downward after curling over at the top is considerably lessened. 
Actually, our shallow water theory cannot be expected to yield a 
good approximation near the breaking point since the curvature of 
the water surface is likely to be large there. However, since the motion 
should be given with good accuracy at points outside the immediate 
vicinity of the breaking point it might be possible to refine the treat- 
ment of the breaking problem along the following lines: consider the 
motion of a fixed portion of the water between a pair of planes located 
some distance in front and in back of the breaking point. The motion 
of the water particles outside the bounding planes can be considered as 
given by our shallow water theory. We might then seek to determine 
the motion of the water between these two planes by making use of a 
refinement of the shallow water theory or by reverting to the original 
exact formulation of the problem in terms of a potential function with 

* In Chapter 12.1 this representation is explained and used to solve other 
problems involving unsteady motions. 



366 



WATER WAVES 




Fig. 10.10.13. Progression and breaking of a wave on a beach of 1 in 10 slope. 

First-order theory 





























'■'■'''?;?//-, 
















• • • > i/J/, 



Fig. 10.10.14. Details of breaking of wave shown in Fig. 10.10.13. First-order 

theory 



LONG WAVES IN SHALLOW WATER 



367 



MEAN LEVEL 





Fig. 10.10.15. Progression and breaking of a wave on a beach of 1 in 10 slope. 

Second-order theory 




Fig. 10.10.16. Details of breaking of wave shown in Fig. 10.10.15. Second-order 

theory 



368 WATER WAVES 

the nonlinear free surface condition and determine it by using finite 
difference methods in a bounded region. 

It is of interest now to return to the problem with which we opened 
the discussion of the present section, i.e. to the problem of a tank with 
a moveable end which is pushed into the water. As we have seen, the 
wave which arises will eventually break. Suppose now we assume 
that the end of the tank continues to move into the water with a uni- 
form velocity. The end result after the initial curling over and break- 
ing will be the creation of a steady progressing wave front which is 
steep and turbulent, behind which the water level is constant and the 



Fig. 10.10.17. The bore in the Tsien Tang River 

water has everywhere the constant velocity imparted to it by the end 
of the tank. Such a steady progressing wave with a steep front is 
called a bore. It is the exact analogue of a steady progressing shock 
wave in a gas. In Figure 10.10.17 we show a photograph, taken from 
the book by Thorade [T.4], of the bore which occurs in the Tsien- 
Tang River as a result of the rising tide, which pushes the water into 
a narrowing estuary at the mouth of the river. The height of this bore 
apparently is as much as 20 to 30 feet. According to the theory pre- 
sented above, this bore should have been preceded by an unsteady 
phase during which the smooth tidal wave entering the estuary first 
curled over and broke. Methods for the treatment of problems in- 
volving the gradual development of a bore in an otherwise smooth 
flow have been worked out by A. Lax [L.5] (see also Whitham [W.12] ). 
We have, so far, used our basic theorv— the nonlinear shallow 



LONG WAVES IN SHALLOW WATER 369 

water theory— to interpret the solutions of only one type of problem, 
i.e. the problem of the change of form of a pulse moving into still 
water of constant depth. The theory, however, can be used to study 
the propagation of a wave over a beach with decreasing depth just 
as well (cf. the author's paper [S.19]), but the calculations are made 
much more difficult because of the fact that no family of straight 
characteristics exists unless the depth is constant. This problem, in 
fact, brings to the fore the difficulties of a computational nature 
which occur in important problems involving the propagation of flood 
waves and other surges in rivers and open channels in general. Such 
problems will be discussed in the next chapter. 

On an actual beach on which waves are breaking, the motion of 
the water, of course, does not consist in the propagation of a single 
pulse into still water, but rather in the occurrence of an approximately 
periodic train of waves. However, experiments indicate that only a 
slight reflection of the wave motion from the shore occurs. The in- 
coming wave energy seems to be destroyed in turbulence due to break- 
ing or to be converted into the energy of flow of the undertow. In 
other words, each wave propagates, to a considerable degree, in a 
manner unaffected by the waves which preceded it. Consequently the 
above treatment of breaking, in which propagation of a wave into still 
water was assumed, should be at least qualitatively reasonable. An- 
other objection to our theory has already been mentioned, i.e. that 
large curvatures of the water surface near the breaking point seem sure 
to make the results inaccurate. Nevertheless, the theory should be 
valid, except near this point, in many cases of waves on sloping bea- 
ches, since the wave lengths are usually at least 10 to 20 or more times 
the depth of the water in the breaker zone, hence the theory presented 
above should certainly yield correct qualitative results and perhaps 
also reasonably accurate quantitative results. 

Waves do not by any means always break in the manner described 
up to this point. In Fig. 10.10.18a, b we show photographs (given to 
the author by Dr. Walter Munk) of waves breaking in a fashion con- 
siderably at variance with the results of the theory presented here. We 
observe that the waves break, in this instance, by curling over slightly at 
the crest, but that the wave remains, as a whole, symmetrical in shape, 
while the theory presented here yields a marked steepening of the wave 
front and a very unsymmetrical shape for the wave at breaking. 

Observation of cases like that shown in Figure 10.10.18 doubtlessly 
led to the formulation of the theory of breaking due to Sverdrup and 



370 



WATER WAVES 




(b) 

Fig. 10.10.18a, b. Waves breaking at crests 

Munk [S.33]; their theory is based on results taken from the study of 
the solitary wave, which has been discussed in the preceding section.* 
The solitary wave is, by definition, a wave of finite amplitude con- 

* An interesting mathematical treatment of breaking phenomena from this 
point of view was given some time ago by Kenlegan and Patterson [K.13J. 



LONG WAVES IN SHALLOW WATER 371 

sisting of a single elevation of such a shape that it can propagate un- 
changed in form. At first sight, this would seem to be a rather curious 
wave form to take as a basis for a discussion of the phenomena of 
breaking, since it is precisely the change in form resulting in breaking 
that is in question. On the other hand, the waves often look as in 
Figure 10.10.18 and do retain, on the whole, a symmetrical shape,* 
with some breaking at the crest. Actually, the situation regarding the 
two different theories of breaking from the mathematical point of 
view is the following, as we can infer from the discussion of the pre- 
ceding section: Both theories are shallow water theories. In fact, as 
Keller [K.6], and Friedrichs and Hyers [F.13], have shown, the theory 
of the solitary wave can be obtained from the approximation of next higher 
order above that used in the present section, if the assumption is made that 
the motion is a steady motion. In other words, the theory used by 
Sverdrup and Munk is a shallow water theory of higher order than 
the theory used in this section, which furnishes in principle the con- 
stant state as the only continuous wave which can propagate un- 
altered in form. On the other hand, the theory presented here makes 




Fig. 10.10.19. Symmetrical waves breaking at crests 

it possible to deal directly with the unsteady motions, while Sverdrup 
and Munk are forced to approximate these motions by a series of 
different steady motions. One could perhaps sum up the whole matter 
by saying that waves break in different ways depending upon the 
individual circumstances (in particular, the depth of the water com- 
pared with the wave length is very important), and the theory which 
should be used to describe the phenomena should be chosen accord- 
ingly. In fact, Figures 10.10.17 and 10.10.18 depicting a bore and 

* Sverdrup and Munk, like the author, assume that, when considering breaking 
phenomena, each wave in a train can be treated with reasonable accuracy as 
though it were uninfluenced by the presence of the others. 



372 



WATER WAVES 



waves breaking only at the crests of otherwise symmetrical waves 
perhaps represent extremes in a whole series of cases which include 
the breaker shown in Figures 10.10.2 and 10.10.3 as an intermediate 
case. Some pertinent observations on this point have been made by 
Mason [M.4]. A theory has been developed by Ursell [U.ll] which 
differs from the theories discussed here and which may well be appro- 
priate in cases not amenable to treatment by the shallow water theory. 
The paper by Hamada [H.2] referred to above should also be men- 
tioned again in this connection. In particular, Fig. 10.10.19, taken 
from that paper, shows waves created in a tank which break by curling 
at the crest but still preserving a symmetrical form. It is interesting 
to observe that the wave length in this case is almost the same as the 
depth of the water. It is also interesting to add that in this case a 
current of air was blown over the water in the direction of travel of 
the waves. Fig. 10.10.20 shows a similar case, but with somewhat 
greater wave length. The two waves were both generated by a wave 




Fig. 10.10.20. Breaking induced by wind action 



making apparatus at the right; the only difference is that a current 
of air was blown from right to left in the case shown by the lower 
photograph. The breaking thus seems due entirely to wind action in 



LONG WAVES IN SHALLOW WATER 



373 



this case. Finally, Fig. 10.10.21 shows two stages in the breaking of a 
wave in shallow water, when marked dissymmetry and the formation 
of what looks like a jet at the summit of the wave are seen to occur. 
It is of interest, historically and otherwise, to refer once more to the 
case of symmetrical waves breaking at their crests. The wave crests 
in such cases are quite sharp, as can be seen in the photograph shown 
in Fig. 10.10.18. Stokes [S.28] long ago gave an argument, based on 
quite reasonable assumptions, that steady progressing waves with an 
angular crest of angle 120° could occur; in fact, this follows almost at 
once from the Bernoulli law at the free surface when the free surface is 
assumed to be a stream line with an angular point. There is another 
fact pertinent to the present discussion, i.e. that the exact theory for 
steady periodic progressing waves of finite amplitude, as developed 
in Chapter 12.2, shows that with increasing amplitude the waves 
flatten more and more in the troughs, but sharpen at the crests. 




Fig. 10.10.21. Breaking of a long wave in shallow water 



In fact, the terms of lowest order in the development of the free surface 
amplitude rj as given by that theory can easily be found; the result is 

rj(cc) = — a cos x + a 2 cos 2x 
for a wave of wave length 2n. Fig. 10.10.22 shows the result of super- 



374 



WATER WAVES 



imposing the second-order term a 2 cos 2x on the wave — a cos x 
which would be given by the linear theory; as one sees, the effect is as 
indicated. It would be a most interesting achievement to show rigor- 
ously that the wave form with a sharp crest of angle 120° is attained 
with increase in amplitude. An interesting approximate treatment of 
the problem has been given by Davies [D.5]. However, the problem 
thus posed is not likely to be easy to solve; certainly the method of 
Levi-Civita as developed in Ch. 12.2 does not yield such a wave form 
since it is shown there that the free surface is analytic. Presumably, 




Fig. 10.10.22. Sharpening of waves at the crest 

any further increase in amplitude would lead to breaking at the crests 
—hence no solutions of the exact problem would exist for amplitudes 
greater than a certain value. 

10.11. Gravity waves in the atmosphere. Simplified version of the 
problem of the motion of cold and warm fronts 

In practically all of this book we assume that the medium in which 
waves propagate is water. It is, however, a notable fact that some 
motions of the atmosphere, such as tidal oscillations due to the same 
cause as the tides in the oceans, i.e. gravitational effects of the sun 
and moon, as well as certain large scale disturbances in the atmosphere 
such as wave disturbances in the prevailing westerlies of the middle 
latitudes, and motions associated with disturbances on certain dis- 
continuity surfaces called fronts, are all phenomena in which the air 
can be treated as a gravitating incompressible fluid. In addition, one 
of the best-founded laws in dynamic meteorology is the hydrostatic 
pressure law, which states that the pressure at any point in the at- 
mosphere is very accurately given by the static weight of the column 
of air above it. When we add that the types of motions enumerated 
above are all such that a typical wave length is large compared with 



LONG WAVES IN SHALLOW WATER 375 

an average thickness (on the basis of an average density, that is) of 
the layer of air over the earth, it becomes clear that these problems fall 
into the general class of problems treated in the present chapter. Of 
course, this means that thermodynamic effects are ignored, and 
with them the ingredients which go to make up the local weather, 
but it seems that these effects can be regarded with a fair approxima- 
tion as small perturbations on the large scale motions in question. 

There are many interesting problems, including very interesting un- 
solved problems, in the theory of tidal oscillations in the atmosphere. 
These problems have been treated at length in the book by Wilkes [W.2] ; 
we shall not attempt to discuss them here. The problems involved in 
studying wave propagation in the prevailing westerlies will also not 
be discussed here, though this interesting theory, for which papers by 
Charney [C.15] and Thompson [T.10] should be consulted, is being 
used as a basis for forecasting the pressure in the atmosphere by nu- 
merical means. In other words, the dynamical theory is being used for 
the first time in meteorology, in conjunction with modern high speed 
digital computing equipment, to predict at least one of the elements 
which enter into the making of weather forecasts. 

In this section we discuss only one class of meteorological problems, 
i.e. motions associated with frontal discontinuities, or, rather, it 
would be better to say that we discuss certain problems in fluid 
dynamics which are in some sense at least rough approximations to the 
actual situations and from which one might hope to learn something 
about the dynamics of frontal motions. The problems to be treated 
here— unlike the problems of the type treated by Charney and 
Thompson referred to above — are such as to fit well with the preceding 
material in this chapter; it was therefore thought worthwhile to in- 
clude them in this book in spite of their somewhat speculative charac- 
ter from the point of view of meteorology. Actually, the idea of using 
methods of the kind described in this chapter for treating certain special 
types of motions in the atmosphere has been explored by a number of 
meteorologists (cf. Abdullah [A.7], Freeman [F.10], Tepper [T.ll]). 

One of the most characteristic features of the motion of the atmos- 
phere in middle latitudes and also one which is of basic importance 
in determining the weather there is the motion of wave-like disturb- 
ances which propagate on a discontinuity surface between a thin 
wedge-shaped layer of cold air on the ground and an overlying layer 
of warmer air. In addition to a temperature discontinuity there is also 
in general a discontinuity in the tangential component of the wind 



376 WATER WAVES 

velocity in the two layers. The study of such phenomena was initiated 
long ago by Bjerknes and Solberg [B.20] and has been continued 
since by many others. In considering wave motions on discontinuity 
surfaces it was natural to begin by considering motions which depart 
so little from some constant steady motion (in which the discontinuity 
surface remains fixed in space) that linearizations can be performed, 
thus bringing the problems into the realm of the classical linear 
mathematical physics. Such studies have led to valuable insights, 
particularly with respect to the question of stability of wave motions 
in relation to the wave length of the perturbations. (The problems 
being linear, the motions in general can be built up as a combination, 
roughly speaking, of simple sine and cosine waves and it is the 
wave length of such components that is meant here, cf. Haurwitz 
[H.5, p. 234].) One conjecture is that the cyclones of the middle lati- 
tudes are initiated because of the occurrence of such unstable waves 
on a discontinuity surface. 

A glance at a weather map, or, still better, an examination of weath- 
er maps over a period of a few days, shows clearly that the wave 
motions on the discontinuity surfaces (which manifest themselves as 
the so-called fronts on the ground) develop amplitudes so rapidly and 
of such a magnitude that a description of the wave motions over a 
period of, say, a day or two, by a linearization seems not feasible with 
any accuracy. The object of the present discussion is to make a first 
step in the direction of a nonlinear theory, based on the exact hydro- 
dynamical equations, for the description of these motions, that can be 
attacked by numerical or other methods. No claim is made that the 
problem is solved here in any general sense. What is done is to start 
with the general hydrodynamical equations and make a series of 
assumptions regarding the flow; in this way a sequence of three non- 
linear problems (we call them Problems I, II, III), each one furnishing 
a consistent and complete mathematical problem, is formulated. 
One can see then the effect of each additional assumption in 
simplifying the mathematical problem. The first two problems result 
from a series of assumptions which would probably be generally 
accepted by meteorologists as reasonable, but unfortunately even 
Problem II is still pretty much unmanageable from the point of view 
of numerical analysis. Further, and more drastic, assumptions lead 
to a still simpler Problem III which is formulated in terms of three 
first order partial differential equations in three dependent and three 
independent variables (as contrasted with eight differential equations 



LONG WAVES IN SHALLOW WATER 377 

in four independent variables in Problem I). The three differential 
equations of Problem III are probably capable of yielding reasonably 
accurate approximations to the frontal motions under consideration, 
but they are still rather difficult to deal with, even numerically, 
principally because they involve three independent variables*: such 
equations are well known to be beyond the scope of even the most 
modern digital computing machines as a rule. Consequently, still 
further simplifying assumptions are made in order to obtain a theory 
capable of yielding some concrete results through calculation. 

At this point, two different approaches to the problem are proposed. 
One of them, by Whitham [W.12], deals rather directly with the 
three differential equations of Problem III. Two of these equations 
are essentially the same as those of the nonlinear shallow water theory 
treated in the preceding sections of this chapter. These two equations 
—which refer to motions in vertical planes — can therefore be inte- 
grated. Afterwards the transverse component of the velocity is found 
by integrating a linear first order partial differential equation. In this 
way a quite reasonable qualitative description of the dynamics of 
frontal motions can be achieved, at least in special cases, which is 
in good agreement with many of the observed phenomena. However, 
this theory has a disadvantage in that it does not permit a complete 
numerical integration because of a peculiar difficulty at cold fronts. 
(The difficulty stems from the fact that a cold front corresponds in 
this theory to what amounts to the propagation of a bore down the 
dry bed of a stream — a mathematical impossibility. If one had a 
means of taking care of turbulence and friction at the ground, it would 
perhaps be possible to overcome this difficulty.) Nevertheless, the 
qualitative agreement with the observed phenomena is an indication 
that the three differential equations furnishing the basic approximate 
theory from which we start— i.e. those of our Problem III — have in 
them the possibility of furnishing reasonable results. 

The author's method (cf. [S.24]) of treating the three basic differ- 
ential equations is quite different from that of Whitham, but it un- 
fortunately involves a further assumption which has the effect of 
limiting the applicability of the theory. The guiding principle was that 

* The work of Freeman [F.9, 10] is based on a theory which could be con- 
sidered as a special case of Problem III in which the Coriolis terms due to the 
rotation of the earth are neglected and the motion is assumed at the outset to 
depend on only one space variable and the time. The idea of deriving the theory 
resulting in Problem III occurred to the author while reading Freeman's paper 
and, indeed, Freeman indicates the desirability of generalizing his theory. 



378 



WATER WAVES 



differential equations in only two independent variables should be 
found, but that the number of dependent variables need not be so 
ruthlessly limited. Finally, it is highly desirable to obtain differential 
equations of hyperbolic type in order that the theory embodied in the 
method of characteristics becomes available in formulating and solv- 
ing concrete problems. These objectives can be attained by making 
quite a few further simplifying assumptions with respect to the me- 
chanics of the situation. The result is what might be called Problem 
IV. The theory formulated in Problem IV is embodied in a system of 
four nonlinear first order partial differential equations of hyperbolic 
type in four dependent and two independent variables. A numerical 
integration of these equations is possible, but the labor of integrating 
the equations is so great that only meagre results are so far available. 
Once Whitham's theory and Problem IV have been formulated, 
one is led once more to consider dealing with Problem III numerically 
in spite of the fact that there are three independent variables in this 
case; in Problem IV, and also in the theory by Whitham, for that 
matter, the basic idea is that variations in the ^/-direction are less 
rapid than those in the ^-direction, and thus a finite difference scheme 
in two space variables and the time might be possible under such 
special circumstances. 




Warm 



Ground 



Fig. 10.11.1. A stationary front 

We proceed to the derivation of the basic approximate theory. To 
begin with, a certain steady motion (called a stationary front) is taken 
as an initial state, and this consists of a uniform flow of two super- 
imposed layers of cold and warm air, as indicated in Figure 10.11.1. 
The z-axis is taken positive upward* and the x, ?/-plane is a tangent 

* Here we deviate from our standard practice of taking the i/-axis as the 
vertical axis, in order to conform to the usual practice in dynamic meteorology. 
This should cause no confusion, since this section can be read to a large extent 
independently of the rest of the book. 



LONG WAVES IN SHALLOW WATER 379 

plane to the earth. The rotation of the earth is to be taken into 
account but, for the sake of simplicity, not its sphericity— a common 
practice in dynamic meteorology. The coordinate system is assumed 
to be rotating about the 2-axis with a constant angular velocity 
Q = co sin 99, with co the angular velocity of the earth and <p the lati- 
tude of the origin of our coordinate system. (The motivation for this is 
that the main effects one cares about are found if the Coriolis terms 
are included, and that neglect of the curvature of the earth has no 
serious qualitative effect.) As indicated in Figure 10.11.1, the cold air 
lies in a wedge under the warm air and the discontinuity surface 
between the two layers is inclined at angle a to the horizontal. The 
term "front" is always applied to the intersection of the discontinuity 
surface with the ground, and in the present case we have therefore as 
initial state a stationary front running along the #-axis. The wind 
velocity in the two layers is parallel to the <z-axis (otherwise the dis- 
continuity surface could not be stationary), but it will in general be 
different in magnitude and perhaps even opposite in direction in the 
two layers. The situation shown in Figure 10.11.1 is not uncommon. 
For instance, the rc-axis might be in the eastward direction, the y-axis 
in the northward direction and the warm air would be moving in the 
direction of the prevailing westerlies. The origin of the cold air at the 
ground is, of course, the cold polar regions. We shall see later that 
such configurations are dynamically correct and that the angle a 
is uniquely determined (and quite small, of the order of J°) once 
the state of the warm air and cold air is given. (The discontinuity 
surface is not horizontal because of the Coriolis force arising from 
the rotation of the earth.) 

We proceed next to describe what is observed to happen in many 
cases once such a stationary front starts moving. In Figure 10.11.2 a 
sequence of diagrammatic sketches is given which indicate in a general 
way what can happen. All of the sketches show the intersection of the 
moving discontinuity surface (cf. Figure 10.11.1) with the ground (the 
x, ?/-plane with the ?/-axis taken northward, the #-axis taken eastward). 
The shaded area indicates the region on the ground covered by cold 
air, while the unshaded region is covered at the ground by warm air. 
Of course, the cold air always lies in a thin wedge under a thick layer 
of warm air. In Figure 10.11.2a the development of a bulge in the 
stationary front toward the north is indicated.* Such a bulge then 

* What agency serves to initiate and to maintain such motions appears to 
be a mystery. Naturally such an important matter has been the subject of a great 

(footnote continued) 



380 



WATER WAVES 



frequently deepens and at the same time propagates eastward with a 
velocity of the order of 500 miles per day. It now becomes possible to 
define the terms cold front and warm, front. As indicated in Figure 10. 
11.2ft, the cold front is that part of the whole front at which cold air is 
taking the place of warm air at the ground, and the warm front is the 




Front 




Warm 
Front 



(0) 



(b) 




(c) (d) 

Fig. 10.11.2. Stages in the motion of a frontal disturbance 



portion of the whole front where cold air is retreating with warm air 
taking its place at the ground. Since such cold and warm fronts are 
accompanied by winds, and by precipitation in various forms — in 
fact, by all of the ingredients that go to make up what one calls 



deal of discussion and speculation, but there seems to be no consistent view about 
it among meteorologists. In applying the theory derived here no attempt is made 
to settle this question a priori: we would simply take our dynamical model, 
assume an initial condition which in effect states that a bulge of the kind just 
described is initiated, and then study the subsequent motion by integrating the 
differential equations subject to appropriate initial and boundary conditions. 
However, if the approximate theory is really valid, such studies might perhaps 
be used, or could be modified, in such a way as to throw some light on this im- 
portant and vexing question. 



LONG WAVES IN SHALLOW WATER 381 

weather— it follows that the weather at a given locality in the middle 
latitudes is largely conditioned by the passage of such frontal dis- 
turbances. Cold fronts and warm fronts behave differently in many 
ways. For example, the cold front in general moves faster than the 
warm front and steepens relative to it, so that an originally symme- 
trical disturbance or wave gradually becomes distorted in the manner 
indicated in Figure 10.11.2c. This process sometimes— though by no 
means always— continues until the greater portion of the cold front 
has overrun the warm front; an occluded front, as indicated in Figure 
10.11.2c?, is then said to occur. The prime object of what follows is to 
derive a theory— or perhaps better, to invent a simplified dynamical 
model— capable of dealing with fluid motions of this type that is not 
on the one hand so crude as to fail to yield at least roughly the observed 
motions, and on the other hand is not impossibly difficult to use 
for the purpose of mathematical discussion and numerical calculation. 
Since it is desired that this section should be as much as possible 
self contained, we do not lean on the basic theory developed earlier 
in this book. Thus we begin with the classical hydrodynamical equa- 
tions. The equations of motion in the Eulerian form are taken: 

du dp 

dt dx 



(10.11.1 



Q "17 = - al + Q F (x) 



dv dp 

dt dy 



Q^= -z7 + Q F (v) 



dw dp 

e*^ -&+<?*« -e* 

with d/dt (the particle derivative) defined by the operator d/d/ + 
u d/dx + v d/dy -f- w d/dz. In these equations u, v, w are the velocity 
components relative to our rotating coordinate system, p is the pres- 
sure, q the density, qF {x) etc. the components of the Coriolis force 
due to the rotation of the coordinate system, and qg is the force of 
gravity (assumed to be constant). These equations hold in both the 
warm air and the cold air, but it is preferable to distinguish the de- 
pendent quantities in the two different layers; this is done here 
throughout, by writing u\ v' 9 w' for the velocity components in the 
warm air and similarly for the other dependent quantities. 

We now introduce an assumption which is commonly made in 
dynamic meteorology in discussing large-scale motions of the atmos- 
phere, i.e. that the air is incompressible. In spite of the fact that such 



382 WATER WAVES 

an assumption rules out thermodynamic processes, it does seem rather 
reasonable since the pressure gradients which operate to create the 
flows of interest to us are quite small and, what is perhaps the decisive 
point, the propagation speed of the disturbances to be studied is very 
small compared with the speed of sound in air (i.e. with disturbances 
governed by compressibility effects). It would be possible to consider 
the atmosphere, though incompressible, to be of variable density. 
However, for the purpose of obtaining as simple a dynamical model as 
possible it seems leasonable to begin with an atmosphere having a 
constant density in each of the two layers. As a consequence of these 
assumptions we have the following equation of continuity: 

(10.11.2) u x + v y + w a = 0. 

The equations (10.11.1) and (10.11.2) together with the conditions 
of continuity of the pressure and of the normal velocity components 
on the discontinuity surface, the condition w = at the ground, 
appropriate initial conditions, etc. doubtlessly yield a mathematical 
problem — call it Problem I— the solution of which would furnish a 
reasonably good approximation to the observed phenomena. Unfor- 
tunately, such a problem is still so difficult as to be far beyond the 
scope of known methods of analysis— including analysis by numerical 
computation. Thus still further simplifications are in order. 

One of the best-founded empirical laws in dynamic meteorology is 
the hydrostatic pressure law, which states that the pressure at any 
point in the atmosphere is very closely equal to the static weight of 
the column of air vertically above it. This is equivalent to saying that 
the vertical acceleration terms and the Coriolis force in the third 
equation of (10.11.1) can be ignored with the result 

dp 

(10.11.3) ■£ =- Q g. 

This is also the basis of the long-wave or shallow water theory of 
surface gravity waves, as was already mentioned above. Since the 
vertical component of the acceleration of the particles is thus ignored, 
it follows on purely kinematical grounds that the horizontal compo- 
nents of the velocity will remain independent of the vertical coordinate 
z for all time if that was the case at the initial instant / = 0. Since we 
do in fact assume an initial motion with such a property, it follows 
that we have 



LONG WAVES IN SHALLOW WATER 383 

(10.11.4) u = u(x, y, t), v = v(x, y, t), w = 0.* 

The first two of the equations of motion (10.11.1) and the equation of 
continuity (10.11.2) therefore reduce to 



(10.11.5) 



1 

+ uu x + vu y = — — p x -f F u 



1 

v t + uv x +vv y = — — p y + F (y) 



where we use subscripts to denote partial derivatives and subscripts 
enclosed in parentheses to indicate components of a vector. The 
Coriolis acceleration terms are now given by 

F {x) = 2co sin cp • v = h) 



(10.11.6) 

1 F iy) = — 2co sin cp - u = — Au 

when use is again made of the fact that w = 0. (The latitude angle cp y 
as was indicated earlier, is assumed to be constant. ) We observe once 
more that all of these relations hold in both the warm and cold layers, 
and we distinguish between the two when necessary by a prime on the 
symbols for quantities in the warm air. It is perhaps also worth men- 
tioning that the equations (10.11.5) with F ix) and F (y) defined by 

(10.11.6) are valid for all orientations of the x, y-axes; thus it is not 
necessary to assume (as we did earlier, for example) that the original 
stationary front runs in the east-west direction. 

We have not so far made full use of the hydrostatic pressure law 
(10.11.3). To this end it is useful to introduce the vertical height 
h = h(x, y, t) of the discontinuity surface between the warm and cold 
layers and the height h! = h'(x, y,t) of the warm layer itself (see 
Figure 10.11.3). Assuming that the pressure p' is zero at the top of 
the warm layer we find by integrating (10.11.3): 

(10.11.7) p'(x, y, z, t) = g'g(h' - z) 

for the pressure at any point in the warm air. In the cold air we have, 
in similar fashion: 

(10.11.8) p(x, y, z, t) = Q 'g(h' -h) + Qg(h - z) 

* It would be wrong, however, to infer that we assume the vertical displacements 
to be zero. This is a peculiarity of the shallow water theory in general which 
results, when a formal perturbation series is used, because of the manner in 
which the independent variables are made to depend on the depth (cf. Ch. 2 and 
early parts of the present chapter). 



384 



WATER WAVES 



when the condition of continuity of pressure, p' == p for z = h, is 
used. (The formula (10.11.8) is the starting point of the paper by 
Freeman [F.10] which was mentioned earlier.) Insertion of (10.11.8) 




Fig. 10.11.3. Vertical height of the two layers 

in (10.11.5) and of (10.11.7) in (10.11.5)' yields the following six 
equations for the six quantities w, v, h, u', v', h': 



+ Xv 



— hi 



(10.11.9) 


U t + UU X +VU y = — g 


J*:+(i- 


e/ J 


(cold air) 


V t + UV X + VV y = — g 


L^'+l 1 - 


-7)'"] 




U X + Vy = 






u t + u'u' x + v'u'y = - gti x + h)' 




(10.11.10) 
(warm air) 


v\ + u'v' x + v'v y = — gti y — Xu' 






u 'x+ v 'v = °- 







These equations together with the kinematic conditions appropriate 
at the surfaces z = h and z = h', and initial conditions at t = 0, 
would again constitute a reasonable mathematical problem— call it 
Problem II— which could be used to study the dynamics of frontal 
motions. The Problem II is much simpler than the Problem I formu- 
lated above in that the number of dependent quantities is reduced 
from eight to six and, probably still more important, the number of 
independent variables is reduced from four to three. These simplifica- 
tions, it should be noted, come about solely as a consequence of assum- 
ing the hydrostatic pressure law, and since meteorologists have much 



LONG WAVES IN SHALLOW WATER 385 

evidence supporting the validity of such an assumption, the Problem 
II should then furnish a reasonable basis for discussing the problem 
of frontal motions. Unfortunately, Problem II is just about as in- 
accessible as Problem I from the point of view of mathematical and 
numerical analysis. Consequently, we make still further hypotheses 
leading to a simpler theory. 

As a preliminary to the formulation of Problem III we write down 
the kinematic free surface conditions at z = h and z = h! (the dyna- 
mical free surface conditions, p = at z = h! and p = p' at z = h, 
have already been used.) These conditions state simply that the 
particle derivatives of the functions z — h(x, y, t) and z — h'(x, y, t) 
vanish, since any particle on the surface z — h — or the surface 
z — ti = remains on it. We have therefore the conditions 

( uh x + vh v + h t = 

(10.11.11) u'h x + v'h y + h t = 
{ u'ti x + v'h' y -\-h\-0, 

in view of the fact that w vanishes everywhere. It is convenient to 
replace the third equations (the continuity equations) in the sets 
(10.11.9) and (10.11.10) by 

(10.11.12) (uh) x + (vh) y +h t = 0, and 

(10.11.13) [u'(h' - h)] x + [v'(h f - h)] y + (h' - h) t = 0, 

which are readily seen to hold because of (10.11.11). In fact, the last 
two equations simply state the continuity conditions for a vertical 
column of air extending (in the cold air) from the ground up to z = h, 
and (in the warm air) from z = h to z = h'. 

We now make a really trenchant assumption, i.e. that the motion 
of the warm air layer is not affected by the motion of the cold air layer. 
This assumption has a rather reasonable physical basis, as might be 
argued in the following way: Imagine the stationary front to have 
developed a bulge in the ^/-direction, say, as in Figure 10.11.4a. The 
warm air can adjust itself to the new condition simply through a 
slight change in its vertical component, without any need for a change 
in u' and v' , the horizontal components. This is indicated in Figure 
10.11.46, which is a vertical section of the air taken along the line 
AB in Figure 10.1 1.4a; in this figure the cold layer is shown with a 
quite small height — which is what one always assumes. Since we 
ignore changes in the vertical velocity components in any case, it thus 
seems reasonable to make our assumption of unaltered flow conditions 



386 



WATER WAVES 



in the warm air. However, in the cold air one sees readily — as indicat- 
ed in Figure 10.11.4c— that quite large changes in the components 
u, v of the velocity in the cold air may be needed when a frontal dis- 
turbance is created. Thus we assume from now on that u' , v' 9 h' have 
for all time the known values they had in the initial steady state in 



,///// 


/ / WCold / 


/airy//// 
II 1 




j////w 






^hl/l/llllmi b 


11111/11/ 


Warm 


air 


Will ML . 



(a) 




Fig. 10.11.4. Flows in warm and cold air layers 

which v' = 0, v! = const. The differential equations for our Problem 
III can now be written as follows: 



(10.11.14) 



U t + UU X + VU y = — g 
V t + UV X -f VUy = — g 

{ h t + (uh) x + (vh) y = 0. 



~i K+ ( 1 - e i) h -] +Xv 

;K + (i - j) *.] '-*• 



LONG WAVES IN SHALLOW WATER 



387 



in which h' x and h' y are known functions given in terms of the initial 
state in the warm air. The initial state, in which v' = v = 0, u = 
const., u' = const., must satisfy the equations (10.11.9) and (10.11.10); 
this leads at once to the conditions 



(10.11.15) 



*; = 


X 








ky = 


X 
— -u - 


-°-K 

Q 


X/Q' 

g\9 


/ -u 


(- 


QJ 


"'(»- 


--) 



h'=0 



/L = 



for the slopes of the free surfaces initially. The slope h y of the dis- 
continuity surface between the two layers is nearly proportional to 
the velocity difference u' — u since q'/q differs only slightly from 
unity, and it is made quite small under the conditions normally en- 
countered because of the factor X, which is a fraction of the angular 
velocity of the earth. The relation for the slope h y of the stationary 
discontinuity surface is an expression of the law of Margules in meteor- 
ology. The differential equations for Problem III can, finally, be ex- 
pressed in the form: 



(10.11.16) 
Problem III 



u t + uu 



vu y +g'(l Jh x = Xv 

v t +uv x + vv y -f g (l \h y = X I — u' — u\ 



\ h t + (uh) x + (vk)y = 0, 



by using the formulas for ti x and ti y given in (10.11.15). We note that 
the influence of the warm air expresses itself through its density q' 
and its velocity u' . The three equations (10.11.16) undoubtedly have 
uniquely determined solutions once the values of u, v, and h are given 
at the initial instant t = 0, together with appropriate boundary con- 
ditions if the domain in x, y is not the whole space, and such solutions 
might reasonably be expected to furnish an approximate descrip- 
tion of the dynamics of frontal motions.* Unfortunately, these equa- 
tions are still quite complicated. They could be integrated numerically 

* These equations are in fact quite similar to the equations for two-dimen- 
sional unsteady motion of a compressible fluid with h playing the role of the 
density of the fluid. 



388 WATER WAVES 

only with great difficulty even with the aid of the most modern high- 
speed digital computers— mostly because there are three independent 
variables. 

Consequently, one casts about for still other possibilities, either of 
specialization or simplification, which might yield a manageable 
theory. One possibility of specialization has already been mentioned: 
if one assumes no Coriolis force and also assumes that the motion is 
independent of the y- coordinate, one obtains the pair of equations 



(• - jY- 



1u t -f uu x + g 
h t + (uh) x = 

which are identical with the equations of the one-dimensional shallow 
water gravity wave theory. These equations contain in them the 
possibility of the development of discontinuous motions— called 
bores in sec. 10.7— and this fact lies at the basis of the discussions by 
Freeman [F.10] and Abdullah [A.7]. In such one-dimensional treat- 
ments, it is clear that it is in principle not possible to deal with the 
bulges on fronts and their deformation in time and space, since such 
problems depend essentially on both space variables x and y. Another 
possibility would be a linearization of the differential equations 
(10.11.16) based on assuming small perturbations of the frontal sur- 
face and of the velocities from the initial uniform state. This procedure 
might be of some interest, since such a formulation would take care of 
the boundary condition at the ground, while the existing linear treat- 
ments of this problem do not. However, our interest here is in a non- 
linear treatment which permits of large displacements of the fronts. 
One such possibility, devised by Whitham [W.12], involves essentially 
the integration of the first and third equations for u and h as functions 
of x and t, regarding y as a parameter, and derivatives with respect to 
y as negligible compared with derivatives with respect to x, and assum- 
ing initial values for v; this is feasible by the method of characteristics. 
Afterwards, v would be determined by integrating the second equa- 
tion considering u and h as known, and this can in principle be done 
because the equation is a linear first order equation under these con- 
ditions. As stated earlier, this procedure furnishes qualitative results 
which agree with observations. In addition, the discussion can be 
carried through explicitly in certain cases, by making use of solutions 
of the type called simple waves, along exactly the same lines as in 
sec. 10.3 above. We turn, therefore, to this first of two proposed 



LONG WAVES IN SHALLOW WATER 389 

approximate treatments of Problem III, as embodied in equations 
(10.11.16). 

The basic fact from which Whitham starts is that the slope a — h y 
of the discontinuity surface is small initially, as we have already seen 
in connection with the second equation of (10.11.15), and the fact that 
X is a fraction of the earth's angular velocity, and is expected to remain 
in general small throughout the motions considered. Since the Coriolis 
forces are of order a also (since they are proportional to A) it seems 
clear that derivatives of all quantities with respect to y will be small of 
a different order from those with respect to x; it is assumed therefore 
that u y , h y and v y are all small of order a, but that u x and h x are finite. 
Furthermore we can expect that the main motion will continue to be 
a motion in the ^-direction, so that the ^-component v of the velocity 
will be small of order a while the ^-component u remains of course 
finite. Under these circumstances, the equations (10.11.16) can be 
replaced by simpler equations through neglect of all but the lowest 
order terms in a in each equation; the result is the set of equations 

h t -\- uh x + hu x = 
(10.11.18) ^ u t ~\~ uu x -\- kh x = 

v t + uv x = — kh y + A I — u' — u\ 
with the constant k defined bv 



(10.11.19; 



(-?)• 



A considerable simplification has been achieved by this process, since 
the variable y enters into the first two equations of (10.11.18) only as 
a parameter and these two equations are identical with the equations 
of the shallow water theory developed in the preceding sections of this 
chapter if k is identified with g and h with r\. This means that the 
theory developed for these equations now becomes available to dis- 
cuss our meteorological problems. Of course, the solutions for h and 
u will depend on the variable y through the agency of initial and 
boundary conditions. Once u(x, y, t) and h(x, y, t) have been obtained, 
they can be inserted in the third equation of (10.11.18), which then 
is a first order linear partial differential equation which, in principle 
at least, can be integrated to obtain v when arbitrary initial conditions 
v = v(x, y, 0) are prescribed. The procedure contemplated can thus be 
summed up as follows : the motion is to be studied first in each vertical 



390 WATER WAVES 

plane y — constant by the same methods as in the shallow water 
theory for two-dimensional motions (which means gas dynamics 
methods for one-dimensional unsteady motions), to be followed by 
a determination of the "cross-component" v of the velocity through 
integration of a first order linear equation which also contains the 
variable y, but only as a parameter. 

This is in principle a feasible program, but it presents problems too 
complicated to be solved in general without using numerical com- 
putations. On the other hand we know from the earlier parts of this 
chapter that interesting special solutions of the first two equations of 
(10.11.18) exist in the form of what were called simple waves, and 
these solutions lend themselves to an easy discussion of a variety of 
motions in an explicit way through the use of the characteristic form 
of the equations. In order to preserve the continuity of the discussion 
it is necessary to repeat here some of the facts about the characteristic 
theory and the theory of simple waves; for details, sees. 10.2 and 10.3 
should be consulted. 

By introducing the new function c 2 = kh, replacing h, we obtain 
instead of the first two equations in (10.11.18) the following equations: 

[ 2c t -f 2uc x 4- cu r = 
(10.11.20) 

{ u t + uu x + 2cc x = 0. 

Thus the quantity c = Vkh, which has the dimensions of a velocity, 
is the propagation speed of small disturbances, or wavelets— in ana- 
logy with the facts derived in sec. 10.2. These equations can in turn 
be written in the form 



Ft +iu±c) r* 



(u ± 2c) = 0, 



which can be interpreted to mean that the quantities u± 2c are con- 
stant along curves C ± in the as, 2-plane such that dxjdt = u ± c: 



(10.11.21) 



dec 
u + 2c = const, along C + : — = u + c 

dx 
u — 2c = const, along C_: — = u — c. 



These relations hold in general for any solutions of (10.11.20). Under 
special circumstances it may happen that u — 2c, for example, has the 
same constant value on all C_ characteristics in a certain region; in 



LONG WAVES IN SHALLOW WATER 391 

that case since u + 2c is constant along each C + characteristic it 
follows that u and c would separately by constant along each of the 
C + characteristics, which means that these curves would all be straight 
lines. Such a region of the flow (the term region here being applied 
with respect to some portion of an x, /-plane) is called a simple wave. 
It is then a very important general fact that any flow region adjacent 
to a region in which the flow is uniform, i.e. in which both c and u are 
everywhere constant (in both space and time, that is), is a simple 
wave, provided that u and c are continuous in the region in question. 

It is reasonable to suppose that simple waves would occur in cases 
of interest to us in our study of the dynamics of frontal motions, 
simply because we do actually begin with a flow in which u and h 
(hence also c) are constant in space and time, and it seems reasonable 
to suppose that disturbances are initiated, not everywhere in the flow 
region, but only in certain areas. In other words, flows adjacent to 
uniform flows would occur in the nature of things. Just how in detail 
initial or boundary conditions, or both, should be prescribed in order 
to conform with what actually occurs in nature is, as has already been 
pointed out, something of a mystery; in fact one of the principal 
objects of the ideas presented here could be to make a comparison of 
calculated motions under prescribed initial and boundary conditions 
with observed motions in the hope of learning something by inference 
concerning the causes for the initiation and development of frontal 
disturbances as seen in nature. 

One fairly obvious and rather reasonable assumption to begin with 
might be that u, v, and h are prescribed at the time t = to have 
values over a certain bounded region of the upper half (y > 0) of 
the x, y-p\ane (cf. Fig. 10.11.1) in such a fashion that they differ from 
the constant values in the original uniform flow with a stationary 
front. According to the approximate theory based on equations (10.11. 
18), this means, in particular, that in each vertical plane y = y — an 
x, /-plane— initial conditions for u(x, y , t) and h(x, y , t) would be 
prescribed over the entire #-axis, but in such a way that u and h are 
constant with values u — u > 0, h = h ^ (hence c = c = 
Vkh )* everywhere except over a certain segment x x ^ x ^ x 2 , as 
indicated in Fig. 10.11.5. The positive characteristics C + are drawn in 
full lines, the characteristics C_ with dashed lines in this diagram, 

* It should, however, always be kept in mind in the discussion to follow 
tat c , pai 
y = const. 



392 



WATER WAVES 



which is to be interpreted as follows. Simple waves exist everywhere 
in the x, Z-plane except in the triangular region bounded by the C + 
characteristic through A and the C_ characteristic through B and 
terminating at point C; in this region the flow could be determined 




Fig. 10.11.5. Simple waves arising from initial conditions 



numerically, for example by the method indicated in sec. 10.2 above 
(in connection with Fig. 10.2.1). The disturbance created over the 
segment AB propagates both "upstream" and "downstream" after a 
certain time in the form of two simple waves, which cover the regions 
bounded by the straight (and parallel) characteristics issuing from 
A, B, and C. In other words the disturbance eventually results in two 
distinct simple waves, one propagating upstream, the other down- 
stream, and separated by a uniform flow identical with the initial 
state. In our diagram it is tacitly assumed that c > | u |, i.e. that the 
flow is subcritical in the terminology of water waves (subsonic in 
gas dynamics)— otherwise no propagation upstream could occur. We 
have supposed u to be positive, i.e. that the ^-component of the flow 
velocity in the cold air layer has the same direction as the velocity in 
the warm air, which in general flows from the west, but it can be (and 
not infrequently is) in the westward rather than the eastward direc- 
tion. Since the observed fronts seem to move almost invariably to the 
eastward, it follows, for example, that it would be the wave moving 
upstream which would be important in the case of a wind to the west- 
ward in the cold layer, and a model of the type considered here — in 
which the disturbance is prescribed by means of an initial condition 



LONG WAVES IN SHALLOW WATER 393 

and the flow is subcritical— implies that the initial disturbances are 
always of such a special character that the downstream wave has a 
negligible amplitude. For a wind to the eastward, the reverse would 
be the case. All of this is, naturally, of an extremely hypothetical 
character, but nevertheless one sees that certain important elements 
pertinent to a discussion of possible motions are put in evidence. 

The last remarks indicate that a model based on such an initial 
disturbance may not be the most appropriate in the majority of cases. 
In fact, such a formulation of the problem is open to an objection 
which is probably rather serious. The objection is that such a motion 
has its origin in an initial impulse, and this provides no mechanism by 
which energy could be constantly fed into the system to "drive" the 
wave. Of course, it would be possible to introduce external body forces 
in various ways to achieve such a purpose, but it is not easy to see how 
to do that in a rational way from the point of view of mechanics. 
Another way to introduce energy into the system would be to feed it 
in through a boundary— in other words formulate appropriate bound- 
ary conditions as well as initial conditions. For the case of fronts 
moving eastward across the United States, a boundary condition 
might be reasonably applied at some point to the east of the high 
mountain system bordering the west coast of the continent, since 
these mountain ranges form a rather effective north-south barrier 
between the motions at the ground on its two sides. In fact, a cold 
front is not infrequently seen running nearly parallel to the moun- 
tains and to the east of them— as though cold air had been deflected 
southward at this barrier. Hence a boundary condition applied at 
some point on the west seems not entirely without reason. In any case, 
we seek models from which knowledge about the dynamics of fronts 
might be obtained, and a model making use of boundary conditions 
should be studied. We suppose, therefore, that a boundary condition 
is applied at x = 0, and that the initial condition for t = 0, x > is 
that the flow is undisturbed, i.e. u = u = const., c = c = const.. 
(Again we remark that we are considering the motion in a definite 
vertical plane y = y .) In this case we would have only a wave propa- 
gating eastward — in effect, we replace the influence of the air to the 
westward by an assumed boundary condition. The general situation 
is indicated in Fig. 10.11.6. There is again a simple wave in the region 
of the x, Z-plane above the straight line x = (u -f c )t which marks 
the boundary between the undisturbed flow and the wave arising 
from disturbances created at x = 0. This is exactly the situation which 



394 



WATER WAVES 



is treated at length in sec. 10.3; in particular, an explicit solution of 
the problem is easily obtained (cf. the discussion in sec. 10.4) for 
arbitrarily prescribed disturbances in the values of either of the two 
quantities u or c. Through various choices of boundary conditions it 
is possible to supply energy to the system in a variety of ways. 




Fig. 10.11.6. Wave arising from conditions applied at a boundary 



We proceed next to discuss qualitatively a few consequences which 
result if it is assumed that frontal disturbances can be described in 
terms of simple waves in all vertical planes y = y = const, at least 
over some ranges in the values of the ^/-coordinate. (We shall see later 
that simple waves are not possible for all values of y. ) In this discussion 
we do not specify hoiv the simple wave was originated— we simply 
assume it to exist. Since we consider only waves moving eastward 
(i.e. in the positive ^-direction) it follows that the straight character- 
istics are C + characteristics, and hence that u — 2c is constant (in 
each plane y = const.) throughout the wave; we have therefore 

(10.11.22) u - 2c = A(y), 

with A(y) fixed by the values u and c (y) in the undisturbed flow: 

(10.11.22)! A(y) = u - 2c (y). 

In addition, as explained before, we know that u -j- 2c is a function of 
y alone on each positive characteristic dx/dt = u + c; hence u and c 
are individually functions of y on each of these characteristics. There- 
fore, the characteristic equation may be integrated to yield 

(10.11.23) x = £ + (u + c)t, 

where £ is the value of x at t = 0. (The time t = should be thought 
of as corresponding to an arbitrary instant at which simple waves 
exist in certain planes y = const. ) Now, the values of u and c on the 



LONG WAVES IN SHALLOW WATER 395 

characteristic given by (10.11.23) are exactly the same as the values 
(for the same value of y) at the point t = 0, x = £; therefore, if we 
suppose, for example, that c is a given function C(x, y) at t = 0, the 
value of c in (10.11.23) is C(£, y) and the value of u is, from (10.11.22), 
A(y) + 2C(£, y). Thus the simple wave solution can be described by 
the equations 

l ' = C(S, y), 
(10.11.24) « = A(y) + 2C(f, y), 

\x = i + {A(y) + 8C(f, y)}t. 

(Although the arbitrary function occurring in a simple wave could be 
specified in other ways, it is convenient for our purposes to give the 
value of h, and hence c, at t = 0.) 

We could write down the solution for the "cross component", or 
north-south component, v of the velocity in this case; by standard 
methods (cf. the report of Whitham [W.12]) it can be obtained by 
integrating the linear first order partial differential equation which 
occurs third in the basic equations (10.11.18). To specify the solution 
of this equation uniquely an initial condition is needed; this might 
reasonably be furnished by the values v = v(x, y) at the time / = 0. 
The result is a rather complicated expression from which not much 
can be said in a general way. One of the weaknesses of the present 
attack on our problem through the use of simple waves now becomes 
apparent: it is necessary to know values of v some time subsequent to 
the initiation of a disturbance in order to predict them for the future. 

It is possible, however, to draw some interesting conclusions from 
the simple wave motions without considering the north-south com- 
ponent of the velocity. For example, suppose we consider a motion 
after a bulge to the northward in an initially stationary front had 
developed as indicated schematically in Fig. 10.11.2. In a plane y = 
const, somewhat to the north of the bulge we could expect the top of 
the cold air layer (the discontinuity surface, that is) as given by h(x, t) 
to appear, for t = say, as indicated in Fig. 10.11.7. The main fea- 
tures of the graph are that there is a depression in the discontinuity 
surface, but that h > so that this surface does not touch the ground. 
(The latter possibility will be discussed later.) Assuming that the 
motion is described as a simple wave, we see from (10.11.24) that the 
value of c = Vkh at the point x — x x is equal to the value of c which 
was at the point x = £ x at t = 0, where ^ 1 = x 1 — {A(y) + SC(i l3 y)}t v 
That is, the value c = C(( l3 y) has been displaced to the right by an 



396 



WATER WAVES 



amount {A(y) + 3C(£ V y)}t v Since this quantity is greater for greater 
values of C, the graph of h becomes distorted in the manner shown in 
Fig. 10.11.7: the "negative region" (where h x < 0) steepens whilst the 
"positive region" (where h x > 0) flattens out. The positive region 
continues to smooth out, but, if the steepening of the negative region 




t = t, > x=x, x 

Fig. 10.11.7. Deformation of the discontinuity surface 



were to continue indefinitely, there would ultimately be more than 
one value of C at the same point, and the wave, as in our discussion 
of water waves (cf. sec. 10.6 and 10.7), starts to break. Clearly the 
latter event occurs when the tangent at a point of the curve in 
Fig. 10.11.7 first becomes vertical. At this time, the continuous solu- 
tion breaks down (since c and u would cease to be single-valued 
functions) and a discontinuous jump in height and velocity must be 
permitted. In terms of the description of the wave by means of the 
characteristics, what happens is that the straight line characteristics 
converge and eventually form a region with a fold. Such a disconti- 
nuous "bore" propagates faster than the wavelets ahead of it (the 
paths of the wavelets in the x, 2-plane are the characteristics) in a 
manner analogous to the propagation of shock waves in gas dynamics 
and bores in water. 

In the above paragraph we supposed that h and c were different 
from zero, and hence the discussion does not apply to the fronts, which 



LONG WAVES IN SHALLOW WATER 397 

are by definition the intersection of the discontinuity surface with 
the ground. When c = there are difficulties, especially at cold 
fronts, but nevertheless a few pertinent observations can be made, 
assuming the motion to be a simple compression wave with u — 2c 
constant. When c = 0, it follows that u = u — 2c , and since c = 
Vkh and h = cny with a the initial inclination of the top of the cold 
air layer, it follows that u = u — 2Vcnky in this case: But u then 
measures the speed of the front itself in the ^-direction, since a particle 
once on the front stays there; consequently for the speed u f of the 
front we have 

(10.11.25) u f = u - 2Voiky. 

Thus the speed of the front decreases with y, and on this basis it 
follows that a northward bulge would become distorted in the fashion 
indicated by Fig. 10.11.8, and this coincides qualitatively with obser- 
vations of actual fronts. 

Actually, things are not quite as simple as this. If c = 0, it follows 
from the first equation of (10.11.20) that c t + uc x = on such a 
locus, and this in turn means that c = on the particle path defined 
by dxjdt = u. At the same time the C + and C_ characteristics have 
the same direction, since they are given by dxjdt = u ±c. On the 
other hand, we have, again from (10.11.20), u t + uu x = — kh x and 
we see that the relation u = const, along a characteristic for which 
c = cannot be satisfied unless h x = 0. In connection with Fig. 
10.11.7 we have seen that the rising portion toward the east of a de- 
pression in the discontinuity surface tends to flatten out, while the 
falling part from the west tends to steepen and break because the 
higher portions tend to move faster and crowd the lower portions. 
Thus when h, and hence c, tends to zero the tendency will be for 
breaking to occur at the cold front, but not at the warm front. The 
slope of the discontinuity surface at the cold front will then be infinite. 
However, a bore in the sense described above cannot occur since there 
must always be a mass flux through a bore: the motion of the cold 
front is analogous to what would happen if a dam were broken and 
water rushed down the dry bed of a stream. Without considering in 
some special way what happens in the turbulent motion caused by such 
continuous breaking at the ground, it is not possible to continue our 
discussion of the motion of a cold front along the present lines, al- 
though such a problem is susceptible to an approximate treatment. 
Nevertheless, this discussion has led in a rational way to a qualitative 



398 



WATER WAVES 



explanation for the well-known fact that a warm front does indeed 
progress in a relatively smooth fashion as compared with the turbu- 
lence which is commonly observed at cold fronts. Thus near a cold 
front the height of the cold air layer may be considerably greater than 
in the vicinity of the warm front, where h £± 0; consequently the speed 
of propagation of the cold front could be expected to be greater than 



Warm 





Fig. 10.11.8. Deformation of a moving front 



near the warm front (as indicated by the dotted modifications of the 
shape of the cold front in Fig. 10.11.8), with the consequence that the 
gap between the two tends to close, and this hints at a possible ex- 
planation for the occlusion process. One might also look at the matter 
in this way: Suppose c ^ 0, but is small in the trough of the wave 
shown in Fig. 10.11.7. If breaking once begins, it is well known that 



LONG WAVES IN SHALLOW WATER 399 

the resulting bore moves with a speed that is greater than the propa- 
gation speed of wavelets in the medium in front (to the right) of it. 
although slower than the propagation speed in the medium behind it. 
Again one sees that the tendency for the wave on the cold front side 
to catch up with the wave on the warm front side is to be expected on 
the basis of the theory presented here. 

Finally we observe that the velocity of the wave near the undisturbed 
stationary front is u , but well to the north it is given roughly by 
u f = u — 2 Vcnky, which is less than u . There is thus a tendency to 
produce what is called in meteorology a cyclonic rotation around the 
center of the wave disturbance. 

To sum up, it seems fair to say that the approximate theory embo- 
died in equations (10.11.18), even when applied to a very special type 
of motions (i.e. simple waves in each plane y — const.), yields a 
variety of results which are at least qualitatively in accord with 
observations of actual fronts in the atmosphere. Among the pheno- 
mena given correctly in a qualitative way are: the change in shape of 
a wave as it progresses eastward, the occurrence of a smooth wave at 
a warm front but a turbulent wave at a cold front, and a tendency to 
produce the type of motion called a cyclone. 

It therefore seems reasonable to suppose that the differential 
equations of our Problem III, which were the starting point of the 
discussion just concluded, contain in them the possibility of dealing 
with motions which have the general characteristics of frontal motions 
in the atmosphere, and that numerical solutions of the equations of 
Problem III might well furnish valuable insights. This is a difficult 
task, as has already been mentioned. However, an approximate theory 
different from that of Whitham is possible, which has the advantage 
that no especial difficulty arises at cold fronts, and which would per- 
mit a numerical treatment. This approximate theory might be con- 
sidered as a new Problem IV. 

The formulation of Problem IV was motivated by the following 
considerations. If one looks at a sequence of weather maps and 
thinks of the wave motion in our thin wedge of cold air, the re- 
semblance to the motion of waves in water which deform into brea- 
kers (especially in the case of frontal disturbances which develop into 
occluded fronts) is very strong. The great difference is that the wave 
motion in water takes place in the vertical plane while the wave mo- 
tion in our thin layer of cold air takes place essentially in the horizon- 
tal plane. When the hydrostatic pressure assumption is made in the 



400 WATER WAVES 

case of water waves the result is a theory in exact analogy to gas 
dynamics, and thus wave motions with an appropriate "sound speed" 
become possible even though the fluid is incompressible— the free 
surface permits the introduction of the depth of the water as a de- 
pendent quantity, this quantity plays the role of the density in gas 
dynamics, and thus a dynamical model in the form of a compressible 
fluid is obtained. It would seem therefore reasonable to try to invent 
a similar theory for frontal motions in the form of a long-wave theory 
suitable for waves which move essentially in the horizontal, rather 
than the vertical, plane, and in which the waves propagate essentially 
parallel to the edge of the original stationary front, i.e. the #-axis. In 
this way one might hope to be rid of the dependence on the variable y 
at right angles to the stationary front, thus reducing the independent 
variables to two, x and t; and if one still could obtain a hyperbolic 
system of differential equations then numerical treatments by finite 
differences would be feasible. This program can, in fact, be carried out 
in such a way as to yield a system of four first order nonlinear differ- 
ential equations in two independent and four dependent variables 
which are of the hyperbolic type. 

Once having decided to obtain a long- wave theory for the horizontal 
plane, the procedure to be followed can be inferred to a large extent 
from what one does in developing the same type of theory for gravity 
waves in water, as we have seen in Chapter 2 and at the beginning of 
the present chapter. To begin with it seems clear that the displace- 
ment ?)(x, t) of the front itself in the ^/-direction should be introduced 
as one of the dependent quantities — all the more since this quantity 
is anyway the most obvious one on the weather maps. To have such a 
"shallow water" theory in the horizontal plane requires— unfortuna- 
tely— a rigid "bottom" somewhere (which is, of course, vertical in 
this case), and this we simply postulate, i.e. we assume that the 
^/-component v of the velocity vanishes for all time on a vertical plane 
y = d = const, parallel to the stationary front along the #-axis (see 
Figure 10.11.9). The velocity v(x,y,t) is then assumed to vary 
linearly* in y, and its value at the front, y = r](x, t), is called v (x, t). 
The intersection of the discontinuity surface z = h(x, y, t) with the 
plane y = d is a curve given by z = h(x, t), and we assume that the 
discontinuity surface is a ruled surface having straight line generators 
running from the front, y = r}(x, t), to the curve z = h(x, t), and 

* The analogous statement holds also in the long-wave theory in water (to 
lowest order in the development parameter, that is). 



LONG WAVES IN SHALLOW WATER 



401 



parallel to the y, s-plane. Finally, we assume (as in the shallow water 
theory) that u, the ^-component of the velocity, depends on x and t 




lU.t) 



Fig. 10.11.9. Notations for Problem IV 



only: u = 
lations 

(10.11.26) 



(10.11.27; 



u(x, t). The effect of these assumptions is to yield the re- 

y - n(x, t) 



h(x, y, t] 



v(x, y, t) 



- y\{x, t) 
6 — y 



h(x, t), 



v(x, t), 



- T}(X, t) 

as one readily sees. In addition, we assume that a particle that is once 
on the front y — rj(x, t) = always remains on it, so that the relation: 

(10.11.28) v(x,t) =rj t +ur] x 

must hold. The four quantities u(x, t), r](x, t), h(x, t), and v(x, t) are 
our new dependent variables. Differential equations for them will be 
obtained by integrating the basic equations (10.11.16) of Problem III 
with respect to y from y = rj to y = d— which can be done since the 
dependence of u, v, and h on y is now explicitly given — and these three 
equations together with (10.11.28) will yield the four equations we 
want. 

Before writing these equations down it should be said that the 
most trenchant assumption made here is the assumption concerning 
the existence of the rigid boundary y = 6. One might think that as 



402 WATER WAVES 

long as the velocity component v dies out with sufficient rapidity in 
the ^/-direction such an assumption would yield a good approximation, 
but the facts in the case of water waves indicate this to be not suffi- 
cient for the accuracy of the approximation: with water waves in very 
deep water the vertical component of the velocity (corresponding to 
our v here) dies out very rapidly in the depth, but it is nevertheless 
essential for a good approximation to assume that the ratio of the 
depth down to a rigid bottom to the wave length is small. However, 
such a rigid vertical barrier to the winds does exist in some cases of 
interest to us in the form of mountain ranges, which are often much 
higher than the top of the cold surface layer (i.e. higher than the 
curve z = h(x, t) in Figure 10.11.9). In any case, severe though this 
restriction is, it still seems to the author to be worth while to study 
the motions which are compatible with it since something about the 
dynamics of frontal motions with large deformations may be learned 
in the process. In particular, one might learn something about the 
kind of perturbations that are necessary to initiate motions of the 
type observed, and under what circumstances the motions can be 
maintained. 

In carrying out the derivation of the differential equations of our 
theory according to the plan outlined above, we calculate first a 
number of integrals. The first of these arise from (10.11.26) and 
(10.11.27): 

r 8 h r d i- 

hdy = (y — rj)dy = -h(d — rj), 

J t] o V J n ■" 



c" v r s i 



-V(d -7]). 



From these we derive by differentiations with respect to x and t an- 
other set of relations: 



1 _ 1_ 

h x dy = -h x (d - rj) — -hrj x , 

1 1 

v x dy = -v x (d - rj) +-vr) x , 



v t dy = -v t (d - rj) + -vrj t , 

1 , 1_ 

h t dy =-h t (6 -rj) - 2 %. 



LONG WAVES IN SHALLOW AVATER 



403 



(In deriving these relations, it is necessary to observe that the lower 
limit r\ is a function of x and t. ) One additional relation is needed, as 
follows: 

>(5 



J 



6 d ( r° \ i _ i _ 

(hu) x dy = — \u\ hdy \ = - (h x u + ^)(<5 - 17) — « Ami/,. 



v — v " v ' 

We now integrate both sides of the equations (10.11.16) with re- 
spect to y from r\ to d, make use of the above integrals, note that 
u = u(x, t) is independent of y, and divide by d — r\. The result is the 
equations 



(10.11.29) 



1 _ 1 kh 

u t +uu x +~ kh x - - - — 



Tj 



Xv, 



v t +uv. 



uh x + hu t 



uvrj x v 
d—rj d—T) 
hu 



Vt 



rj d 
h 



2kh I g' \ 

1—7] \ g J 



r, 



r\x + fh 



n 



rj t = 0, 



with k a constant replacing the quantity g(l — g'/g). These equations, 
together with (10.11.28), form a system of four partial differential 
equations for the four functions u, tj, v, and It. By analogy with gas 
dynamics and the nonlinear shallow water theory, it is convenient to 
introduce a new dependent quantity c (which will turn out to be the 
propagation speed of wavelets) through the relation 



(10.11.30) 



i a -i4 -■?)*■ 



The quantity c is real if g' is less than g, and this holds since the warm 
air is lighter than the cold air. In terms of this new quantity the 
equations (10.11.28) and (10.11.29) take the form 

1 
u t + uu x + 2cc x — -7], 



(10.11.31 



d — tj 



AV, 



2 

4c 2 



v t + uv. 



2c t + cu x + 2uc x = 



2X 



( w -H 



CO 



d — tj' 



Vt + urj. 



V. 



It is now easy to write the equations (10.11.31) in the characteristic 



404 WATER WAVES 

form simply by replacing the first and third equations by their sum 
and by their difference. The result is: 

c 
u t + (u+c)u x +2 {c t + (u+c)c x }— {ri t + (u+c)ri x }=\Xv, 



(10.11.32 



c 
u t +(u—c)u x -2{c t +(u—c)c x }+- {rj t +(u—c)rj x }=^v, 



4>c 2 

Vt +uv x = — — 2X 

0—7] 



(u- e -u'), 



As one sees, the equations are in characteristic form: the characteristic 
curves satisfy the differential equations 

dx dx dx 

(10.11.83) Tt =u+c, Tt =u-c, - = u, 

and each of the equations (10.11.32) contains only derivatives in the 
direction of one of these curves. The characteristic curves defined by 
dx/dt = u are taken twice. Thus one sees that the quantity c is indeed 
entitled to be called a propagation speed, and small disturbances can 
be expected to propagate with this speed in both directions relative to 
the stream of velocity u. (In the theory by Whitham, in which the 
motion in each vertical plane y = const, is treated separately, the 
propagation or sound speed of small disturbances is given by Vkh. 
The sound speed in the theory given here thus represents a kind of 
average with respect to y of the sound speeds of Whitham's theory.) 
Since the propagation speed depends on the height of the disconti- 
nuity surface, it is clear that the possibility of motions leading to 
breaking is inherent in this theory. 

Once the dynamical equations have been formulated in character- 
istic form it becomes possible to see rather easily what sort of subsi- 
diary initial and boundary conditions are reasonable. In fact, there 
are many possibilities in this respect. One such possibility is the 
following. At time t = it is assumed that u = const., r\ = 0, h = 
const, (as in a stationary front), but that r\ t = f(x) over a segment of 
the «r-axis. In other words, it is assumed that a transverse impulse is 
given to the stationary front over a portion of its length. The sub- 
sequent motion is uniquely determined and can be calculated nume- 
rically. Another possibility is to prescribe a stationary front at t = 



LONG WAVES IN SHALLOW WATER 405 

for x > 0, say, and then to give the values of all dependent quantities* 
at x = as arbitrary functions of the time; i.e. to prescribe a boundary 
condition which allows energy to be introduced gradually into the 
system. One might visualize this case as one in which, for example, 
cold air is being added or withdrawn at a particular point (x = in 
the present case). This again yields a problem with a uniquely deter- 
mined solution, and various possibilities are being explored numeri- 
cally. 

It was stated above that the most objectionable feature of the 
present theory is the assumption of a fixed vertical barrier back of the 
front. There is, however, a different way of looking at the problem as 
a whole which may mitigate this restriction somewhat. One might try 
to consider the motion of the entire cap of cold air that lies over the 
polar region, using polar coordinates (d, cp) (with 6 the latitude angle, 
say). One might then consider motions once more that depend 
essentially only on <p and t by getting rid of the dependence on 6 
through use of the same type of assumptions (linear behavior in 6, 
say ) as above. Here the North Pole itself would take the place of the 
vertical barrier (v = 0!). The result is again a system of nonlinear 
equations— this time with variable coefficients. Of course, it would 
be necessary to begin with a stationary flow in which the motion takes 
place along the parallels of latitude. 

All in all, the ideas presented here would seem to yield theories 
flexible enough to permit a good deal of freedom with regard to initial 
and other conditions, so that one might hope to gain some insight into 
the complicated dynamics of frontal motions by carrying out numeri- 
cal solutions in well-chosen special cases. 

10.12. Supercritical steady flows in two dimensions. Flow around bends. 
Aerodynamic applications 

The title of this section is a slight misnomer, since the flows in 
question are really three-dimensional in nature; however, since we 
consider them here only in terms of the shallow water theory, the 
depth dimension is left out. Thus the velocity is characterized by the 
two components (u, w) in the horizontal plane (the x, z-plane); and 
these quantities, together with the depth h of the water at any point 
constitute the quantities to be determined in any given problem. By 

* In the numerical cases so far considered we have had | c | < | u | so that 
even on the f-axis all four dependent quantities can be prescribed. 



406 WATER WAVES 

specializing the general equations (2.4.18), (2.4.19), (2.4.20), of the 
shallow water theory as derived in Chapter 2 for the case of a steady 
flow, the differential equations relevant for this section result. They 
can also be derived readily from first principles, as follows: Assuming 
that the hydrostatic pressure law holds and that the fluid starts from 
rest (or any other motion in which the vertical component of the 
velocity of the water is zero) it follows that the vertical component of 
the velocity remains zero and that u and w are independent of the ver- 
tical coordinate. The law of continuity can thus be readily derived for 
a vertical column; for a steady flow it is 

(10.12.1) (hu) x + (hw) z = 0. 

We assume that the flows we study are irrotational, and hence that 

(10.12.2) u z — w x = 0. 

The Bernoulli law then holds and can be written in the form 

(10.12.3) (u 2 + w 2 ) + 2gh = const. 

In these equations h is the depth of the water at any point. By using 

(10.12.3) to express h in terms of u and w, and introducing the quan- 
tity c by the relation 

(10.12.4) c 2 = gh 
we obtain the equation 

(10.12.5) (c 2 — u 2 )u x — uw(w x + u z ) + (c 2 — w 2 )w z = 0, 

and this equation together with (10.12.2), with c defined in terms of 
u and w through (10.12.4) and (10.12.3), constitute a pair of first 
order partial differential equations for the determination of u(x, z) 
and w(x, z). 

The theory of these latter equations can be developed, as in the 
cases treated previously in this chapter, by using the method of 
characteristics, provided that the quantity c remains always less than 
the flow speed everywhere, i.e. provided that 

(10.12.6) c 2 < u 2 + w 2 . 

The flow is then said to be supercritical. (In hydraulics the contrast 
subcritical — supercritical is commonly expressed as tranquil-shoot- 
ing.) Only then do real characteristics exist. We shall not develop 
this theory here, but rather indicate some of the problems which have 
been treated by using the theory. Complete expositions of the char- 



LONG WAVES IN SHALLOW WATER 407 

acteristic theory can be found in the paper by Preiswerk [P. 16], and 
in Chapter IV of the book by Courant and Friedrichs [C.9]. The theory 
is, of course, again perfectly analogous to the theory of steady two- 
dimensional supersonic flows in gas dynamics. 




Fig. 10.12.1. Hydraulic jump 

We have already encountered an interesting example of a flow 
which is in part supercritical, in part subcritical, i.e. the case of a 
hydraulic jump in which the character of the flow changes on passage 
through the discontinuity. Figure 10.12.1 is a photograph, taken 
from the paper of Preiswerk, of such a hydraulic jump. Figure 10.12.2, 
also taken from the paper of Preiswerk, shows a more complicated 
case in which hydraulic jumps occur at oblique angles to the direction 
of the flow. The picture shows a flow through a sluice in a dam, with 
conditions (i.e. depth differences above and below the dam) such that 
supercritical flow develops in the sluice, and changes in level take place 
so abruptly that they might well be treated as discontinuities (as was 
done in earlier sections in the treatment of bores). The two disconti- 
nuities at the sides of the sluice (marked 1 and 2 in the figure) are 
turned toward each other and eventually intersect to form a still 



408 



WATER WAVES 



higher one (marked 1 + 2). Such oblique discontinuities can be 
treated mathematically; the details can be found in the works cited 
above. 

Another interesting problem of the category considered here is the 




Fig. 10.12.2. Hydraulic jumps at oblique angles to the direction of the flow 



problem of supercritical flow around a bend in a stream. This type of 
problem is relevant not only for flows in water, but also for certain 
flows in the atmosphere (for which see Freeman [F.9]). It is possible 
in these cases to have flows of the type which are mathematically of 
the kind called simple waves in earlier sections. This means that one 
of the families of characteristics is a set of straight lines along each of 
which wand w (hence also h) are constant. Even the notion of a cen- 
tered simple wave can be realized in these cases. Suppose that the 
flow comes with constant supercritical velocity along a straight wall 



LONG WAVES IN SHALLOW WATER 409 

(cf. Fig. 10.12.3) until a smooth bend begins at point A. The straight 
characteristics are denoted by C + in the figure; they form a set of 




Fig. 10.12.3. Supercritical flow around a smooth bend 

parallel lines in the region of constant flow, which then terminates 
along the C + characteristic through the point A, where the bend be- 
gins; beyond that characteristic a variable regime begins. The straight 
characteristics themselves are called Mach lines; they have physical 
significance and would be visible to the eye: the Mach lines are lines 
along which infinitesimal disturbances of a supercritical flow are 
propagated; in an actual flow they would be made visible because of 
the existence of small irregularities on the surface of the wall of 
the bend. If the bend contracts into a sharp corner, the straight 
characteristics, or Mach lines, which lie in the region in which the 
flow is variable, all emanate from the corner, as indicated in Fig. 
10.12.4; the flow as a whole consists of two different uniform flows 




Fig. 10.12.4. Supercritical flow around a sharp corner 

connected through a centered simple wave. If the bend in the stream 
is concave toward the flow, rather than convex as in the preceding 
two cases, the circumstances are quite different, since the Mach lines 
would now converge, rather than diverge, in certain portions of the 
flow, as indicated in Fig. 10.12.5. Overlapping of the characteristics 
would mean mathematically that the depth and velocity would be 
multi-valued at some points in the flow; this being physically impos- 



410 



WATER WAVES 



sible it is to be expected that something new happens and, in fact, 
the development of a hydraulic jump is to be expected. If the bend 




Fig. 10.12.5. Mach lines for a supercritical flow around a concave bend 

is a sharp angle, as in Fig. 10.12.6, the configuration consisting of two 
uniform flows parallel to the walls of the bend and connected by an 
oblique hydraulic jump is mathematically possible, and it occurs in 
practice. 

Having considered flows delimited on one side only by a wall, it is 
natural to consider next flows between two walls as in a sluice or 
channel of variable breadth. (Such flows are analogous to two-dimen- 




Fig. 10.12.6. Oblique hydraulic jump 



sional steady flows through nozzles in gas dynamics. ) The possibilities 
here are very numerous, and most of them lead to cases not describ- 
able solely in terms of simple waves. They are of considerable import- 
ance in practice. For example, v. Karman [K.2] was led to the study 
of particular flows of this type because of their occurrence in bends in 
the concrete spillways designed to carry the flows of the Los Angeles 






LONG WAVES IN SHALLOW WATER 



411 



river basin through the city of Los Angeles; the seasonal rainfall is so 
heavy and the terrain so steep that supercritical flows are the rule 
rather than the exception during the rainy season. Experiments for 
sluices of special form were carried out by Preiswerk; Fig. 10.12.7, 

a) 





theoretisch 

gemessen bei h =31,1 mm 

Fig. 10.12.7. Laval nozzle a) Mach lines b) contour lines of the water surface 



for example, shows the result of an experiment in a particular case. 
The upper figure shows the Mach lines, the lower figure shows the 
contour lines of the water surface as given by the theory as well as by 
experiment; as one sees, the agreement is quite good. 

Finally, we discuss briefly some applications of interest because of 
their connection with aerodynamics. Because of the analogy of the 
shallow water theory with compressible gas dynamics, it is of course 
possible to interpret experiments with flows in shallow water in terms 
of the analogous flows in gases. Since it is much cheaper and simpler 



412 



WATER WAVES 



to obtain supercritical flows experimentally in water than it is to 
obtain supersonic flows in gases, it follows that "water table" experi- 
ments (as they are sometimes called) may have considerable import- 
ance for those whose principle interest is in aerodynamics. There is a 
considerable literature devoted to this subject; we mention, for exam- 
ple, papers by Crossley [C.12], Einstein and Baird [E.5], Harleman 




Fig. 10.12.8. Photogram of hydraulic-jump intersection 



[H.3], Laitone [L.l], Bruman [B.19]. Figure 10.12.8 is a photograph, 
taken from the paper by Crossley, showing the interaction of two 
hydraulic jumps; this is a case essentially the same as that shown by 
Fig. 10.12.2. The ripples with short wave lengths constitute an effect 
due to surface tension, and the discontinuities are smoothed out so that 
a hydraulic jump does not really occur; the changes in depth are quite 
abrupt, however. Another important case that has been studied by 
means of the hydraulic analogy is, as a matter of course, the flow 



LONG WAVES IN SHALLOW WATER 



413 



pattern which results when a rigid body (simulating a projectile or an 
airfoil) is immersed in a stream. Figure 10.12.9 shows a photograph of 




Fig. 10.12.9. Shock wave in front of a projectile 




Fig. 10.12.10. Flow pattern of a projectile 



such a flow (taken from the paper by Laitone). The shock wave in 
front of the projectile is well shown. Figure 10.12.10 is another photo- 
graph made by Preiswerk; here, Mach lines are clearly visible. 



414 



WATER WAVES 



10.13. Linear shallow water theory. Tides. Seiches. Oscillations in 
harbors. Floating breakwaters 

Up to now in this chapter we have considered problems of wave 
motion in water sufficiently shallow to permit of an approximation in 
terms of what we call the shallow water theory. This theory is non- 
linear in character, and consequently presents difficulties which are 
often quite formidable. By making the assumption that the wave 
amplitudes in the motions under study are small in addition to the 
assumption that the water is shallow, it is possible to obtain a theory 
which is linear — and thus attackable by many known methods — 
and which is also applicable with good approximation in a variety 
of interesting physical situations. We begin by deriving the linear 
shallow water theory under conditions sufficiently general to permit 
us to discuss the cases indicated in the heading of this section. (A 
brief mention of the linear shallow water theory was made in Chapter 
2 and in Chapter 10.1.) 

The linear shallow water theory could of course be derived by 
appropriate linearizations of the nonlinear shallow water theory. It is, 
however, more convenient — and perhaps also interesting from the 
standpoint of method— to proceed by linearizing first the basic general 
theory as developed in Chapter 1, and afterwards making the ap- 
proximations arising from the assumption that the water is shallow. 
In other words, we shall begin with the exact linear theory of Chapter 
2.1, and proceed to derive the linear shallow water theory from it. One 
of the advantages of this procedure is that the error terms involved 
in the shallow water approximation can be exhibited explicitly. 

We suppose the water to fill a region lying above a fixed surface 



Y(x,z;t) 





Fig. 10.13.1. Linear shallow water theory 

(the bottom) y — — h(x, z), and beneath a surface y = Y(x, z; t), 
the motion of which is for the time being supposed known (cf. Fig. 



LONG WAVES IN SHALLOW WATER 415 

10.13.1). The y-axis is taken vertically upward, and the x, s-plane is 
horizontal. The upper surface of the water given by y = Y(x, z; t) 
will consist partly of the free surface (to be determined, for example, 
by the condition that the pressure vanishes there) and partly of the 
surfaces of immersed bodies; it is, however, not necessary to specify 
more about this surface for the present than that it should represent 
always a small displacement from a rest position of equilibrium of the 
combined system consisting of water and immersed bodies. 

We recapitulate the equations of the exact linear theory as derived 
in Chapter 2.1. The velocity components are determined as the deri- 
vatives of the velocity potential 0(x, y, z;t) which satisfies the Laplace 
equation 

(10.13.1) XX +0 yy +0 ZZ = O 

in the space filled by the water. It is legitimate to assume that all 
boundary conditions at the upper surface of the water are to be satis- 
fied at the equilibrium position; this position is supposed given by 

(10.13.2) y = fj(x, z). 

(The bar over the quantity r\ points to the fact that fj could also be 
interpreted as the average position of the water in the important 
special case in which the motion is a simple harmonic motion in the 
time. ) The x, 2-plane is taken in the undisturbed position of the free 
surface, and this in turn means that fj in (10.13.2) has the value zero 
there. Under any immersed bodies the value of fj will be fixed by the 
static equilibrium position of the given bodies. Thus fj is in all cases 
a given function of x and z; for a floating rigid body, for example, it 
would be determined by hydrostatics. 

The condition to be satisfied at the upper surface is the kinematic 
condition: 

(10.13.3) X Y X + Z Y Z -0 y + Y t = O, 

which states that a particle once on the surface remains on it. At the 
bottom surface, the condition to be satisfied is 

(10.13.3)! x h x + z h z +0 y = O at y = - h(x, z). 

Bernoulli's law for determining the pressure at any point in the water is 

(10.13.4) - + t +gy-W=O. 
Q 

Here we have assumed that there mav be other external forces beside 



416 WATER WAVES 

gravity, and these forces are assumed to be determined by a work 
function W(x, y, z; t) whose space derivatives furnish the force com- 
ponents; in this case it is known that the motion can be irrotational 
and that Bernoulli's law holds in the above form (cf. the derivations 
in Chapter 1 ). We now write the equation of the moving upper surface 
in the form 

(10.13.5) y = Y(x, z; t) = fj(x, z) + rj(x, z; t) 

and assume in accordance with our statement above that r)(x, z; t) re- 
presents a small vertical displacement from the equilibrium position 
given by y = fj. Upon insertion in (10.13.3) and (10.13.4) we find 
after ignoring quadratic terms in r\ and and their derivatives: 

(10.13.6) x fj x + Q z fj z -0 y +rj t =O \ 

p \ at y = rj(x,z) 

(10.13.7) - + & t + g(y + v) - w = ° J 

as boundary conditions to be satisfied at the equilibrium position of 
the upper surface of the water. At points corresponding to a free sur- 
face where p = we would have, for example, fj = and hence 

(10.13.8) -0 y +r] t = O 



, at v = 0. 
(10.13.9) t + gr) — W=0 J y 

A special case might be that in which the motion of a portion of the 
upper surface is prescribed, i.e. r)(x, z; t) as well as fj would be pre- 
sumed known; in such a case the condition (10.13.6) alone would 
suffice as a boundary condition for the harmonic function 0. In some 
of the problems to be treated here, however, we do not wish to assume 
that the motion of some immersed body, for example, is known in 
advance; rather, it is to be determined by the interaction with the 
water which exerts a pressure p(x, z; t) on it in accordance with (10. 
13.7). Thus the exact formulation of our problems would require 
the determination of a harmonic function 0{x, y, z; t) in the space 
between y = — h(x, z) and y = fj which satisfies the conditions (10. 
13.6) and (10.13.7) at the upper surface (in particular the conditions 
(10.13.8) and (10.13.9) on the free surface) and (10.13.3)! at the 
bottom. Additional conditions where immersed bodies occur (to be 
obtained from the appropriate dynamical conditions for such bodies) 
would be necessary to determine the pressure p, which provides the 
"coupling" between the water on the one hand and the immersed 
bodies on the other. Finally, appropriate initial conditions for the 



LONG WAVES IN SHALLOW WATER 417 

water and the immersed bodies at the initial instant would be needed 
if one were to study non-steady motions, or — as will be the case here 
—conditions at oo of the radiation type would be needed if simple 
harmonic motions (that is, steady vibrations instead of transients) 
are studied. It need hardly be said that the difficulties of carrying 
out the solutions of such problems are very great indeed ( cf . Chapter 
9, for example) — so much so that we turn to an approximate theory 
which is based on the assumption that the depth of the water is 
sufficiently small and that the immersed bodies are rather flat.* 

In the derivation of the shallow water theory we start from the 
Laplace equation (10.13.1) for and integrate it with respect to y 
from the bottom to the equilibrium position** of the top surface 
y = fj(x, z) to obtain, after integration by parts: 

(10.13.10) \ V _ h ®yydy = y - y = - j V _ h (0 XX + 0„)dy 

+ h x x +h z z . 

Here, and in what follows, a bar over the quantity means that it is 
to be evaluated at the equilibrium position of the upper surface of the 
water, i.e. for y = fj(x, z), and a bar under the quantity means that it 
is to be evaluated at the bottom y = — h(x, z). From the kinematic 
surface condition (10.13.6) and the condition (10.13.3)! at the bottom, 
we have therefore (due regard being paid to the fact that a bar should 
now be put over in (10.13.6) and under in (10.13.3jj): 



(10.13.11) *=-£j 



®x dy - jr\ Z dy. 

-h oz J -h 



This condition — really a continuity condition — expresses the fact 
that the water is incompressible. Consider next the result of integrat- 
ing by parts the right hand side of (10.13.11); in particular: 

(10.13.12) p_ h x dy = fj0 x + h0 x - \\ h y0 xy dy. 
Since we have 

* In the course of the derivation the terms neglected are given explicitly so 
that a precise statement about them can be made. 

** One sees readily that carrying out the integration to y = rj rather than 
to y = tj + t] yields the same results within terms of second order in small 
quantities. 



418 WATER WAVES 

(10.13.13) p_ h h& xy dy = h® x - h& x 
we may eliminate X from (10.13.12) to obtain: 

(10.13.14) J*^ X dy=(fj + h)0 x - j"_ h (h + y)0 xy dy. 

Indeed, we have quite generally for any function F(cc, y, z; t) the 
formula: 

(10.13.14)! p_ h Fdy=(fj + h)F - \\ (h + y)F y dy. 

Making use of the analogous expression for the integral of Z we 
obtain from (10.13.11) the relation 

(10.13.15) rj t =- [(fj + h)0 x ] x - [(fj + h)0 z ] z +I X +J Z 
in which 

(10.13.16) I =j T1 _ h (h+y)0 xy dy, J = f_ h (h + y)0 ty dy. 

In addition, we have from (10.13.10) in combination with (10.13.14) 
the condition: 

(10.13.17) y = - (fj + h)[0 xx + ZZ ] - h x x - h z z +I X + J 2 , 

as one can readily verify. 

Up to this point we have made no approximations other than those 
arising from linearizing. The essential step in obtaining our approxi- 
mate theory is now taken in neglecting the terms I x and J z . This in 
turn is justified if it is assumed that certain second and third deriva- 
tives of are bounded when h is small and that fj and its first deriva- 
tives are small* of the same order as h: one sees that the terms I x 
and J z in the right hand sides of (10.13.15) and (10.13.17) are then of 
order h 2 while the remaining terms are of order h. Under the free sur- 
face in the case of a simple harmonic oscillation one can show that this 
approximation requires the depth to be small in comparison with the 
wave length. 

Upon differentiating the relation (10.13.7) for the pressure at the 
upper surface of the water with respect to t (again noting that a bar 
should be placed over the term t in (10.13.7)) and using (10.13.15) 
we find the equation 

* This means that the theory developed here applies to immersed bodies 
which are flat. 



LONG WAVES IN SHALLOW WATER 419 

(10.13.18) g*«+f--g W *= fW +*)*.]■ + [(9+*)S.L 

after dropping the terms T^ and J z . This is the basic differential equa- 
tion for the function 0(x, z; t) which holds everywhere on the upper 
surface of the water. In particular, we have at the free surface where 
p = and fj = the equation 

(10.13.19) (h0 x ) x + (h0 z ) z --0 tt =- W t . 

6 S 

We recall that W(x, y, z; t) represents the work function for any- 
external forces in addition to gravity (tide generating forces, for 
example), so that W, its value on the free surface, would be given by 
W(x, 0, z; t). If, in addition, it is assumed that h is a constant, i.e. that 
the depth of the water is uniform, and that gravity is the only external 
force, we would have the equation 

(10.13.19), XX +0 zz -±^ tt = 0, 

that is, the linear wave equation in the two space variables x, z and 
the time t. As a consequence, all disturbances propagate in such a 
case with the constant speed c = Vgh, as is well known for this 
equation. 

If there is an immersed object in the water, the equation (10.13.19) 
holds everywhere in the x, 2-plane exterior to the curve C which 
defines the water line on the immersed body in its equilibrium posi- 
tion. The curve C is supposed given by the equations 

(10.13.20) x = x(s), z = z{s) 

in terms of a parameter s. We must have boundary, or perhaps it is 
better to say, transition conditions at the curve C which connect the 
solutions of (10.13.19) in the exterior of C in an appropriate manner 
with the motion of the water under the immersed body. Reasonable 
conditions for this purpose can be obtained from the laws of conser- 
vation of mass and energy. In deriving these conditions we assume 
W = 0, since we wish to deal only with gravity as the external force 
when considering problems involving immersed bodies. Consider an 
element of length ds of the curve C representing the water line of the 
immersed body (cf. Fig. 10.13.2). The expression 



420 



WATER WAVES 



represents the mass flux through a vertical strip having the normal n 




C(x(s),y(s)) 



Fig. 10.13.2. Boundary at water line of an immersed body 

and extending from the bottom to the top of the water. From (10.13. 
14) x applied for F = n we have 

(10.13.21 ) q j V _ h n dy = Q (fj + h)0 n - Q j V _ h (h + y)0 ny dy 

— Q(V + h)0 n 

where the second term is ignored because it is of order h 2 . Thus it 
would be reasonable to require that (fj + h)0 n should be continuous 
on C since this is the same as requiring that the mass of the water is con- 
served within terms of the order retained otherwise in our theory. For 
the flux of energy across a vertical strip with the normal n we have 

(10.13.22) {J*^ p0 n dy } ds = {- q j V _ h &t®n dy - gg f_ h y0 n dy] ds 

upon making use of the Bernoulli law (10.13.4) for the pressure p 
(when W = 0). Once more we may ignore the second term in the 
brackets since it is of order h 2 . Upon applying (10.13.14 ) x with 
F = t n and again ignoring a term of order h 2 we find 

(10.13.23) [j n _ h p0 n dy)ds = {- Q(fj+h)0 n t } ds. 

Since we have already required that (fj + h)0 n should be continuous, 
we see that the additional requirement, t continuous, ensures the 
continuity of the energy flux. 

As reasonable transition conditions on the curve C delimiting the 
immersed body at its water line we have therefore 



(10.13.24 



(fj + h)0 n , t continuous on C. 



LONG WAVES IN SHALLOW WATER 421 

Of course, if fj is continuous (e.g. if the sides of the body in contact 
with the water do not extend vertically below the undisturbed free 
surface) it follows that n would then be continuous. 

In order to make further progress it would be necessary to specify 
the properties of the immersed body. However, we have succeeded in 
obtaining the equation (10.13.18) which is generally valid and of basic 
importance for our theory together with the transition conditions 
(10.13.24) valid at the edge of immersed bodies; in particular, we have 
the definitive equation for the free surface itself in the form of the 
linear wave equation (10.13.19). The idea behind this method of ap- 
proximation is to get rid of the depth variable by an integration over 
the depth so that the problems then are considered only in the x, z- 
plane. As a result, the problems are no longer problems in potential 
theory in three space variables, but rather problems involving the wave 
equation with only two space variables, and hence they are more open 
to attack by known methods. How this comes about will be seen in 
special cases in the following. 

As a first example of the application of the above theory we con- 
sider briefly the problem of the tides in the oceans, with a view to 
indicating where this theory fits into the theory of gravity waves in 
general, but not with the purpose of giving a detailed exposition. (For 
details, the long Chapter VIII in Lamb [L.3] should be consulted.) 
To begin with, it might seem incredible at first sight that the shallow 
water theory could possibly be accurate for the oceans, since depths of 
five miles or more occur. However, it is the depth in relation to the 
wave length of the motions under consideration which is relevant. 
The tides are forced oscillations caused by the tide-generating at- 
tractions of the moon and the sun, and hence have the same periods 
as the motions of the sun and moon relative to the earth. These periods 
are measured in hours, and consequently the tidal motions in the 
water result in waves having wave lengths of hundreds of miles;* 
the depth-wave length ratio is thus quite small and the shallow 
water theory should be amply accurate to describe the tides. This 
means, in effect, that the differential equation (10.13.19), or rather, its 
analogue for the case of water lying on a rotating spheroid (with Cori- 
olis terms put in if a coordinate system rotating with the earth is 

* For example, in water of depth 10,000 feet (perhaps a fairly reasonable 
average value for the depth of the oceans) a steady progressing wave having 
a length of 10,000 feet has a period of only 44.2 sec. (cf. Lamb [L.3], p. 369). 
Since the wave length varies as the square of the period, the correctness of our 
statement is obvious. 



422 WATER WAVES 

used), would serve as a basis for calculating tidal motions. Of course, 
the function W t would be defined in terms of the gravitational forces 
due to the attraction of the sun and moon. The variable depth of the 
water in the oceans would come into play, as well as boundary condi- 
tions at the shore lines of the continents. Presumably, W t would be 
analyzed into its harmonic components (which could be obtained 
from astronomical data), the response to each such harmonic would 
be calculated, and the results superimposed. Such a problem consti- 
tutes a linear vibration problem of classical type — it is essentially the 
same as the problem of transverse forced oscillations of a tightly 
stretched non-uniform membrane with an irregular boundary. If it 
were not for one essential difficulty, to be mentioned in a moment, 
such a problem would in all likelihood be solvable numerically by 
using modern high speed computational equipment. The difficulty 
mentioned was pointed out to the author in a conversation with 
H. Jeffreys, and it is that there are difficulties in prescribing an ap- 
propriate boundary condition in coastal regions where there is dissi- 
pation of energy in the tidal motions (in the bay of Fundy, for exam- 
ple, to take what is probably an extreme case). At other coastal re- 
gions the correct boundary condition would of course often be simply 
that the component of the velocity normal to the coast line vanishes. 
Of course, there would also be a difficulty in using a differential 
equation like (10.13.19) near any shores where h = 0, since the dif- 
ferential equation becomes singular at such points. Nevertheless, a 
computation of the tides on a dynamical basis would seem to be a 
worthwhile problem— perhaps it could be managed in such a way as 
to help, in conjunction with observations of the actual tides, in 
providing information concerning the dissipation of energy in such 
motions. 

These remarks might be taken to imply that the dynamical theory 
is not at present used to compute the tides. This is not entirely correct, 
since the tide tables for predicting the tides in various parts of the 
world are based on fundamental consequences of the assumption that 
the tides are indeed governed by a differential equation of the same 
general type as (10.13.19). The point is that the oceans are regarded as 
a linear vibrating system under forced oscillations due to excitation 
from the periodic forces of attraction of the sun and moon. It is 
assumed that all free vibrations of the oceans were long ago damped 
out, and hence, as remarked above, that the tidal motions now exist- 
ing in the oceans are a superposition of simple harmonic oscillations 



LONG WAVES IN SHALLOW WATER 423 

having periods which are very accurately known from astronomical 
observations. To obtain tide tables for any given point a superposition 
of oscillations of these frequencies is taken with undetermined ampli- 
tudes and phases which are then fixed by comparing them with a 
harmonic analysis of actual tidal observations made at the point in 
question. The tide predictions are then made by using the result of 
such a calculation to prepare tables for future times. The dynamical 
theory is thus used only in a qualitative way. An interesting addition- 
al point might be mentioned, i.e. that tides of observable amplitudes 
are sometimes measured which have as frequency the sums or differ- 
ences (or also other linear combinations with integers) of certain of 
the astronomical frequencies, which means from the point of view of 
vibration theory that observable nonlinear effects must be present. 
Another type of phenomenon in nature which can be treated by 
the theory derived here concerns periodic motions of rather long 
period, called seiches, which occur in lakes in various parts of the 
world. The first observations of this kind seem to have been made by 
Forel [F.7] in the lake at Geneva in Switzerland, in which oscillations 
having a period of the order of an hour and amplitudes of up to six 
feet have been observed. In larger lakes still larger periods of oscilla- 
tion are observed — about fifteen hours in Lake Erie, for example. 
A rather destructive oscillation, generally supposed to be of the type 
of a seiche, occurred in Lake Michigan in June 1954; a wave with an 
amplitude of the order of ten feet occurred and swept away a number 
of people who were fishing from piers and breakwaters. What the 
mechanism is that gives rise to seiches in lakes has been the object of 
considerable discussion, but it seems rather clear that the motions 
represent free vibrations of the water in a lake which are excited by 
external forces of an impulsive character, the most likely type arising 
from sudden differences in atmospheric pressure over various portions 
of the water surface. Bouasse [B.15, p. 158] reports, however, that 
the Lisbon earthquake of 1755 caused oscillations in Loch Lomond 
with a period of about 5 minutes and amplitudes of several feet. 
In any case, the periods observed seem to correspond to those calcu- 
lated on the basis of the linear shallow water theory, which should be 
quite accurate for the study of seiches because of their long periods and 
small amplitudes. It follows, therefore, that the differential equation 
(10.13.18) is applicable; we suppose that W t = (since tidal forces 
play no role in this case), and also set fj = since there are no immersed 
bodies to be considered. The differential equation for &(x r z;t) is thus 



424 WATER WAVES 

(10.13.25) (h0 x ) x + (h0 z ) z --0 tt =^. 

The free natural vibrations of the lake are investigated by setting 
p t = and 0(x, z; t) = <p(x, z)e iat in (10.13.25) with the result 

a 2 

(10.13.26) (htp x ) x + (h<p z ) z + -99 = 0. 

As boundary condition along the shore of the lake we would have 

(10.13.27) <p n = 0. 

The problem thus posed is one of the classical eigenvalue problems of 
mathematical physics. Solutions 99 other than the trivial solution 
cp = of (10.13.26) under the homogeneous boundary condition 
(10.13.27) are wanted; such solutions exist only for special values of 
the circular frequency a, and these values yield the natural frequencies 
corresponding to the natural modes cp(x, z) which are correlated with 
them. In general, an infinite set of such natural frequencies occurs. 
For particular shapes and depths h— rectangular or circular lakes of 
constant depth, for example — it is possible to solve such problems 
more or less explicitly. In practice however, lakes have such irregular 
outlines and depths that the determination of the natural frequencies 
and modes requires numerical computation. A reasonable and gener- 
ally applicable method of carrying out such computations is furnished 
here, as in other instances in this and the subsequent chapter, by 
the method of finite differences.* In this method, the derivatives in 
the differential equation and boundary conditions are replaced by 
difference quotients defined by means of the values of the function 
at the discrete points of a net in the domain of the independent varia- 
bles. The resulting finite equations are then solved to yield approxi- 
mate values for the unknown function at the net points. The difference 
approximation will be more accurate for a closer spacing of the net 
points. We proceed to illustrate the method for the case of a lake of 
constant depth in the form of a square of length I on each side, with 
a view to comparing the result with the exact solution which is easy 
to write down in this case. The differential equation (10.13.26) can 
be written in the form 

* A different method was used by Chrystal [C.2] to calculate the periods of 
the free oscillations of Loch Earn; he found good agreement with the obser- 
vations for the first six modes of oscillation. 



LONG WAVES IN SHALLOW WATER 



425 



(10.13.26)! <p xx + (p zz + m 2 cp = 0, 



a 2 /gh 



in this case. A division of the square in a mesh with mesh width 
S = 1/7 is taken, as indicated in Fig. 10.13.3. In numbering the net 
points it has been assumed that only modes of oscillation that are 
symmetrical with respect to the center lines parallel to the sides and to 
to the diagonals are sought, which, however, is not the case for the 











8 9 


y 


4 


7 


9 


9 






3 


6 


8 


9 8 






2 


5 


6 


7 6 




i 


I 


2 


3 


4 




i 


1 


2 




X 














i 













Fig. 10.13.3. Finite differences for a seiche 



mode having the lowest frequency. The boundary condition cp n = is 
satisfied approximately by supposing that the solution is reflected 
over the boundaries in such a way as to yield values which are equal at 
the mirror images in the boundaries, as is also indicated in Fig. 10.13.3. 
The formulas used for approximating the derivatives are defined as 
follows (cf. Fig. 10.13.3): 



tym+l, n 



7i 



dx — 26 

d^P^ - 2( Pm,n + <Pm+l,n + 9?m-l, n 

dcc 2 ~ " S 2 



dcp 
dz 2 



(Pm, n+1 Vm, n-1 



2<Pm, n +<P: 



m, n+1 



+ <Pr 



Consequently, the differential equation (10.13.26)! is replaced at each 
net point (m , n) by the difference equation 

(10.13.28) ~^Cp m ,n+(Pm,n + l+q ) m + l,n+^m,n-l^ m -l,n+^^ 2 q) rnjn = 0. 

Such an equation is written for each of the net points in Fig. 10.13.3. 
The results for points 1,6,9, for example, are: 




426 WATER WAVES 

- 2<p x + 2<p 2 + (<5m)Vi = 

- 499 6 + 993 + (p 7 + 9? 8 + (p 5 + (dm) 2 <p 6 = 

- 4999 + 2<p 8 + <p 7 + <p 10 + {dm)*(p 9 = 0. 

These homogeneous linear equations of course have always the solu- 
tion tp i = 0, i = 1, 2, . . ., 10 unless their determinant vanishes, and 
this condition is a tenth degree equation in the quantity (dm) 2 , the 
smallest root of which furnishes an approximation to the lowest 
frequency. The exact solution of the differential equation (10.13.26) 1 
which satisfies the boundary condition is, in the present case, cp = 
A cos (kjix/l) cos (jnz/l), with k and ; any integers, provided that 
m 2 = n 2 {k 2 + j 2 )/l 2 - A numerical comparison of the lowest value of 
m for the mode having double symmetry — i.e. the value for k = 1, 
;' = 1 — with the value computed from the determinantal equation 
shows the approximate value of m to be too low by 6.5 %. 

However, this mode corresponds to one of the higher eigenvalues, 
so that the accuracy of the finite difference method is rather good. 
The error for the lowest mode is very much smaller, but because of the 
lack of symmetries the amount of calculation needed to determine 
the corresponding frequency would be much greater for the present 
case. If one were to treat a long narrow lake, the calculation would 
be simpler. It could also be advantageous to employ the Rayleigh-Ritz 
method. In principle, similar calculations could be made in more 
complicated cases (for many examples of problems solved along these 
lines see the book by Southwell [S.14]). 

Wave motions in harbors are often of a type suitable for discussion 
in terms of the linear shallow water theory: they are indeed often of 
the type called seiches above. In these cases oscillations of the water 
in the harbor are also commonly excited by the motion at the harbor 
mouth, which in its turn is due, of course, to wave motions generated 
in the open sea. An experimental and theoretical investigation of such 
waves in a model has been carried out by McNown [M.7]. The model 
was in the form of a circle 3.2 meters in diameter with vertical walls. 
The depth of the water in this idealized harbor was 16 cm. An opening 
of angle n/S radians in the harbor wall permitted a connection with a 
large tank in which waves (simulating the open sea) were produced. 
Figures 10.13.4 and 10.13.5 are photographs of the model (taken from 
the paper by McNown), which also show two specific cases of symme- 
trical oscillations. The free vibrations again are governed by equation 
(10.13.26). (It might be noted that McNown makes use of the exact 



LONG WAVES IN SHALLOW WATER 427 

linear theory rather than the shallow water theory. The only difference 
is that the relation between o 2 and m is a 2 = gm tanh mh, instead of 
a 2 = ghm 2 , as given above: the differential equation for the velocity 
potential 0(cc, y, z; t) in the exact linear theory treated in Part I is 




Fig. 10.13.4. and 10.13.5. Waves in a harbor model 

written in the form = A cosh m(y + h)e iat <p(x, z), and <p(x, z) then 
satisfies V 2 (p + m 2 cp = 0. ) Solutions of the differential equation are 
sought in the form 

(10.13.30) (p(r, 6) = J n (mr) cos nd 

in polar coordinates (r, 6), under the assumption that the port is 

closed, i.e. that its boundary is the whole circle r = R. As is well 



428 



WATER WAVES 



known, (p(r, 6) is a solution of (10.13.26)! only if J n (mr) is a Bessel 
function of order n, and since it is reasonable to look only for solutions 
that are bounded we choose the Bessel functions of the first kind 
which are regular at the origin. The boundary condition requires that 

dcp 
dr 



(10.13.31) 



~= 



at r 



R 



and this in turn leads to the condition dJJdr = for r = R. For 
each n this transcendental equation has infinitely many roots m[ n) , 
each corresponding to a mode of oscillation with various nodal dia- 
meters and circles, and with a definite frequency which is fixed by 
a 2 = gm^ (or, more accurately, by a 2 = gm[ n) tanh hmP ). Figure 
10.13.6, obtained by McNown, shows a comparison of observed and 
calculated amplitudes for two modes of oscillation; the upper curve 
is drawn for a motion having no diametral nodes and two nodal circles, 
while the lower is for a motion having two nodal diameters and one 



-2 
-3 
-4 

















to 


















Q 
















■■>, 


K; 


r 












, *s 


K N 


k 




















Jo 


(krj 







































entrance 



Observed 
Theoretical 



center 




Fig. 10.13.6. Comparison of results of experiment and theory for resonant move- 
ments in a circular port 



LONG WAVES IN SHALLOW WATER 



429 



nodal circle. The motions were excited by making waves in the tank, 
and providing an opening for communication with the harbor, as 
noted above. The figures were drawn assuming that the amplitudes 
would agree at the entrance to the harbor — the experimental check 




Fig. 10.13.7. Model of a harbor without breakwater 




Fig. 10.13.8. Model of a harbor with breakwater 

thus applies only to the shapes of the curves. As one sees, the ex- 
perimental and theoretical values are remarkably close. The ampli- 
tudes used were large enough so that nonlinear effects were observed: 
the troughs are flatter than the crests by measurable amounts. Of 
course, having an opening in the harbor wall violates the boundary 
condition assumed, but this effect apparently is slight: changing the 
angle of the opening at the harbor mouth had practically no effect on 



430 



WATER WAVES 



the waves produced, and, in addition, it was found that very little 
wave energy radiates outward through the harbor entrance. 

Problems of harbor design, involving construction of breakwaters, 
location of docks, etc. are commonly studied by constructing models. 
Figs. 10.13.7 and 10.13.8 show two photographs of a model of a har- 
bor,* the first before a breakwater was constructed, the second 
afterward. As one sees, the breakwater has a quite noticeable effect. 
Fig. 10.13.9. shows the same model, with the waves approaching the 
harbor mouth at a different angle, however; as one sees the break- 




Fig. 10.13.9. Model of a harbor with breakwater 

water seems to be on the whole less effective when the wave fronts 
are less oblique to the breakwater. The diamond-shaped pattern, due 
to reflection, of the waves on the sea side of the breakwater is inter- 



7777ZZ2ZZL 



'//////// /A 



h 



777777777777777X77777777777777777 



Fig. 10.13.10. Floating plane slab 



* These photographs were given to the author by the Hydrodynamics Labora- 
tory at California Institute of Technology. 



LONG WAVES IN SHALLOW WATER 431 

esting. Model studies are rather expensive, and consequently it might 
well be reasonable to explore the possibilities of numerical solution 
of the problems, perhaps by using appropriate modifications of the 
method of finite differences outlined above for a simple case. 

We turn next to a discussion of the effect of floating bodies on 
waves in shallow water, on the basis of the theory presented in this 
section. Only two-dimensional motions will be considered (so that all 
quantities are independent of the variable z). The first case to be 
studied is that of the motion of a floating rigid body in the form of a 
thin plane slab (cf. Fig. 10.13.10) in water of uniform depth. Such 
problems have been treated by F. John [J. 5]. The ends of the slab 
are at x = ± a. In accordance with the theory presented above we 
must determine the surface value 0(x, t) and the displacement r](x, t) 
of the board from the differential equations (cf. (10.13.19), (10.13.15) 
with fj = 0, and dropping I x and J z ): 

(10.13.32) XX = —0 W \x\>a 

(10.13.33) rj t =—h0 xx , \x\<a. 

We have dropped the bar over the quantity 0. We have also assumed 
that fj(x) for \x\ < a, the rest position of equilibrium of the board, is 
zero; this is an approximation that is justified because we assume 
that the board is so light that it does not sink appreciably below the 
water surface when in equilibrium. (This assumption is by no means 
necessary — it would not be difficult to deal with the problem if this 
simplifying assumption were not made.) 

Since fj is zero, it follows (cf. (10.13.24)) that the transition con- 
ditions at the ends of the board are 

(10.13.34) X , t continuous at x = ± a. 

We are interested in the problem of the effectiveness of the floating 
board as a barrier to a train of waves coming from the right (x = 
-j- oo ). The equation (10.13.32) has as its general solution 

0(x, t) = F{x - ct) + G(x + ct), c = V~gh 
in terms of two arbitrary functions F and G (as one can readily verify) 
which clearly represent a superposition of two progressing waves 
moving to the right and to the left, respectively, with the speed Vgh. 
It is natural, in our present problem, to expect that for x > a there 
would exist in general both an incoming and an outgoing wave be- 
cause of reflection from the barrier, while for x < — a we would pre- 



432 WATER WAVES 

scribe only a wave going outward (i.e. to the left). We shall see that 

these qualitative requirements lead to a unique solution of our problem. 

We consider only simple harmonic waves; it is thus natural to write 

(10.13.35) 0{x, t) = cp(x)e iat , \ x \ > a, 

(10.13.36) 7](x, t) = v(x)e iat , — a < x < a, 

with the stipulation that the real part is to be taken at the end. (It is 
necessary also to permit cp(x) and v(x) to be complex- valued func- 
tions of the real variable x.) The conditions (10.13.32) and (10.13.33) 
now become 



d 2 w a 2 

(10.13.37) — *- + — <p = 0, \x\>a 
dx 2 gh 

d 2 w ia 

(10.13.38) cte 2+ J V = 0, 1^1 <«• 

The equation (10.13.37) has as general solution 

(10.13.39) (p(x) = Ae~ ikx + Be ikx 
with k given by 

(10.13.40) k = o/Vgh. 
For 0(x, t) we have therefore 

(10.13.41) 0(x, t) = Ae~ i{kx - at) + Be i{kx+at) , 

the first term representing a progressing wave moving to the right, 
the second a wave moving to the left. In our problems we prescribe 
the incoming wave from the right, and hence for cp(x) we write 

(10.13.42) <p(x) = Be ikx + Re~ ikx , x > a, 

in which B is prescribed, while R— the amplitude of the reflected 
wave (more precisely, | R \ is its amplitude) — is to be determined. 
At the left we write 

(10.13.43) (p(x) = Te ikx , 

with T— the amplitude of the transmitted wave — to be determined. 
To complete the formulation of the problem it is necessary to con- 
sider the dynamics of our floating rigid body. We shall treat two cases: 
a) the board is held rigidly fixed in a horizontal position, b) the board 
floats freely in the water. 

a) Rigidly Fixed Board. 

If the board is rigidly fixed we have rj(x,t) =0, and hence (cf. 
(10.13.36)) v(x) = 0. It follows from (10.13.38) that (p xx vanishes 



LONG WAVES IN SHALLOW WATER 



433 



identically under the board and hence that (p(x) is a linear function: 

(10.13.44) tp(x)=yx+d, — a < x < + «• 

Since & x (x, t) furnishes the horizontal velocity component of the 
water, it follows from (10.13.44) and (10.13.35) that the velocity under 
the board is given by ye iat , i.e. it is constant everywhere under the 
board at each instant — a not unexpected result. 

We now write down the transition conditions at x = ±a from 
(10.13.34), making use of (10.13.35) and of (10.13.42) at x = + a 
and (10.13.43) at x = — a; the result is: 



10.13.45) 



Be ika _j_ Jfe-ite 

Be ika __ R e -ika 

Te -ika 

Te -ika 



ya + 6 

yjik 

— ya + d 

yjik. 

Once the real number B— which fixes the amplitude of the incoming 
wave — has been prescribed, these four equations serve to fix the con- 
stants R, T, y, and d and hence the functions @(x, t) and r)(x, t). The 
pressure under the board can then be determined (cf. the expression 
(10.13.4) for Bernoulli's law) from 

(10.13.46) p(x, t) = — Q& t = — Qio(p(x)e iat . 

(Observe that the quantity y in (10.13.4) is zero in the present case.) 
In terms of the dimensionless parameter 

(10.13.47) d = 2fl/A, 

the ratio of the length of the board to the wave length X on the free 
surface, given by (cf. (10.13.41)) 

(10.13.48) X = 27i/k, 
the solution of (10.13.45) is 

OmBe 2d7li 
dm + 1 

Be 26ni 



(10.13.49) 



R 



dm' + 1 
_ dni Be dni 
a Oni + 1 

d = Be Qni . 



The reflection and transmission coefficients are obtained at once: 



434 WATER WAVES 



(10.13.50 



R 



B 



dn 



Vl + 2 7Z 2 



c t 



\T\ 1 



B Vl . + d 2 7l 2 



They depend only upon the ratio 6 = 2a/X, as one would expect. They 
also satisfy the relation C\ + Cf — 1, as they should: this is an ex- 
pression of the fact that the incoming and outgoing energies are the 
same. The following table gives a few specific values for these co- 
efficients : 



d C t 



0.5 0.54 0.85 

1.0 0.30 0.95 

2.0 0.157 0.986 



Thus a fixed board whose length is half the incoming wave length has 
the effect of reducing the amplitude behind it by about 50 percent and 
of reflecting about 72 percent of the incoming energy. One should, 
however, remember that the theory is only for long waves in shallow 
water, and, in addition, it seems likely that the length of the board 
will also play a role in determining the accuracy of the approximation. 
This question has been investigated by Wells [W.10] by deriving the 
shallow water theory in such a way as to include all terms of third order 
in the depth h and studying the magnitude of the neglected terms in 
special cases; in particular, the present case of a floating rigid body 
is investigated. Wells finds that if h/X is small and if a\X (the ratio of 
the half-length of the board to the wave length) is not smaller than 1, 
the neglected higher order terms are indeed negligible, but if ajX is less 
than 1, the higher order terms need not be small. In other words, 
floating obstacles ought to have lengths of the order of the wave 
length of the incoming waves if the shallow water theory to lowest 
order in h is expected to furnish a good approximation. 

It is of interest to study the pressure variation under the board. 
This is given in the present case (cf. (10.13.46)) by 

(10.13.51) p(x, t) = - iagiyx + d)e iot , 

the real part only to be taken. Thus the pressure varies linearly in x, 
but it is a different linear function at different times since y and d are 



LONG WAVES IN SHALLOW WATER 



435 



complex constants. The result of taking the real part of the right hand 
side of (10.13.51) can be readily put in the form 
(10.13.52) p(x, t) = p ± (x) cos at — p 2 ( aj ) sm at 

with 

(10.13.53) 

and 

(10.13.54) 



p x {x) = oqB^^x) sin r + b 2 (x) cos r) 
p 2 {x) = GQB(b 2 (x) sin r — b x (x) cos r) 



b ± (x) 

b 2 (x) 



1 +r> a 



+ 1 



1 +r 2 a 




On 



p 


, 






400 


<rt = 3tt/2^ 






200 




"""crT" 


= 7 7T/4 












-I -5 5 x/0 I 

Fig. 10.13.11. Pressure variations for a stationary board. 6 = 1, /> in pounds I (ft)* 



436 



WATER WAVES 



We have assumed in making these calculations, as stated above, that 
B, which represents the amplitude of the incoming wave, is a real 
number. 

In Fig. 10.13.11 the results of computations for the pressure distri- 
bution for time intervals of 1/4 cycle over the full period are given for 
a special numerical case in which the parameter 6 has the value = 1, 
i.e. the length of the board is the same as the wave length. One ob- 
serves that the pressure variation is greater at the right end than at 
the left, which is not surprising since the board has a damping effect on 
the waves. One observes also that the pressure is sometimes less than 
atmospheric (i.e. it is negative at times, while p = is the assumed 
pressure at the free surface). 

b) Freely Floating Board. 

In Fig. 10.13.12 the notation for the present case is indicated: 
u(t), v(t) represent the coordinates of the center of gravity of the 
board in the displaced position, and co(t) the angular displacement. 
As before, we consider only simple harmonic oscillations and thus 
take u, v, and co in the form 



(10.13.55 



xe l 



ye' 



we 1 




Fig. 10.13.12. A freely floating board 



in which x, y, and w are constants representing the complex ampli- 
tudes of these components of the oscillation. For r\(x, t) we have, 
therefore 



(10.13.56 



f]{x, t) 



x)w]e. 

,iot 



[y + (b 

= (y + xw)e l 

when terms of first order in x, y, and w only are considered. (The hori- 
zontal component of the oscillation is thus seen to yield only a second 
order effect.) The relation (10.13.56) now yields (cf. (10.13.38)): 



(10.13.57 



<Po 



IG 



(y 



xw . 



LONG WAVES IN SHALLOW WATER 



437 



in which <p is the complex amplitude of the velocity potential 0(x, t) = 
q>(x)e iat . Hence op is the following cubic polynomial: 

ig [wx z ya} 2 \ 

Q0 t — Qgrj we have in the 



10.13.58) 



<p(x) 



Since the pressure is given by p = 
present case 

(10.13.59) p(x) = [— iGQ(p(x) — og(y + xw)]e iat . 

The transition conditions (10.13.34) at x = J- a now lead, in the 
same way as above from (10.13.42) and (10.13.43), to the equations 



(10.13.60; 



Be tka + Re 



Be ika - Re 



Te 



Te 



-ika 



■ika 



-ika 



ig /wa 3 
~h \~6~ 


+ ?)+«, 


g /wa 2 
kh\2~ 


+ §a )-j 


Ig I wa 3 ya 2 \ 


g (wa 2 


- y a ) - i 



ay + o 



These four equations are not sufficient to determine the six constants 
R, T, w, y, y, and d. We must make use of the dynamical equations of 
motion of the floating rigid body for this purpose. We have the 
equations of motion: 

(10.13.61) F = Mb, and L = Iol 

at our disposal. In the first equation F and M are the total vertical 

force on the board and its mass, per unit width, and v is the vertical 
acceleration of its center of gravity, / the moment of inertia. L the 
torque, and a the angular acceleration. These dynamical conditions 
then yield the following relations: 

/•o .. pa 

(10.13.62) p dx = Mv, pxdx = lib, 

J —a J —a 

and these in turn lead to the equations 



(10.13.63 



f a ( . T ig(wx 3 yx 2 \ ."I 

I lGQ \~ ~h\ 6" + V) +r ^ -Qgiy+xw 



i: 



IGQ 



h\ 

IG/IVX 



dx 



Mg< 



+ 



y^-\+yx 2 +'6x 



Qg(xy-\-x 2 w) \ dx=^—Iohv. 



438 WATER WAVES 

In the first equation we have ignored the weight of the board, since 
it is balanced by the hydrostatic pressure. The equations (10.13.60) 
and (10.13.63) now determine all of the unknown complex amplitu- 
des. 

We omit the details of the calculations, which can be found in the 
paper by Fleishman [F.5]. In Fig. 10.13.13 the results of calculations 
for the pressure distribution in a numerical case are given. The para- 
meters were chosen as follows: 

= 1, h = 1 ft, B = 1 ft 2 /sec, a = 4 ft, 

M = 18.72 pounds J ft, o = 4.46 radjsec, X = 8 ft. 

It might be added that the value chosen for M is such that the struc- 
ture sinks down 0.0375 feet when in equilibrium. 

A few observations should be made. First of all, we note that in both 
cases the pressure variation at the right end (x = + a), where the 
incoming wave is incident, is greater than at the left end. This is to 
be expected, since the barrier exercises a damping effect on the wave 
going under it. The pressure distribution in the case of the floating 
board is quadratic in x, in contrast with the case of the fixed board in 
which the distribution of pressure was linear in x. Next, we note that 
the pressure variation near the right end of the stationary board is 
greater than at the same end of the floating one; this too might be 
expected since the fixed board receives the full impact of the incident 
wave, while the floating one yields somewhat. Finally, we see that at 
the left end the opposite effect occurs: there the pressure variation 
under the stationary barrier is less than that under the floating barrier. 
This is not surprising either, since the fixed board should damp the 
wave more successfully than the movable board. 

Finally, we take up the case of a floating elastic beam (cf. [F.5]). 
The beam is assumed to extend from x = — I to x = and, as in the 
above cases, to be in simple harmonic motion due to an incoming 
wave from x = + °°- The basic relations for 0(x, t) on the free sur- 
face, and for rj(x, t) under the beam are the same as before: 

(10.13.64) XX = - tt , x>0, x < - I, 

(10.13.65) 7] t = — h& xx , - I <x <0. 

We assume once more that the beam sinks very little below the water 
surface when in equilibrium (i.e. very little in relation to the depth of 



LONG WAVES IN SHALLOW WATER 



439 




P' 
200 




■200 

500 



crt = t/2 




at -- lir/4 



-.5 



5 



p 


L 




400 
200 




a t = 5tt/4>X/ 





SS -SS»* P t = V 






— ~—_ 




■?C\Ci 







x/a 



P- 
400 

200 



-200 



. 












at 


= 3 7T/2/ 






at = 7-rr/A 






r 










L . r ... 


—*~ 



-I -5 

Fig. 10.13.13. Pressure variations for floating board. 6 = 1, p in pounds j (ft)'' 



440 WATER WAVES 

the water), so that the coefficient of @ xx in (10.13.65) can be taken as 
h rather than (h -{- rj) (cf. (10.13.15)), and also the transition condi- 
tions at the ends of the beam are 

I. 



iat 



(10.13.66) 


0,. and t 


continuous 


at x = 


-o, 


x = - 


After writing 












(10.13.67) 


0(x 


,t) 


= cp(x)e iat , 


7](X, 


t) = 


v(x)e' 


we find, as 


before: 












(10.13.68) 


<Pxx + 


a 2 


(p = 0, sc 


' > 0, 


X 


< — 


(10.13.69) 


<Pxx + 


ia 


v = 0, 


- 1 < x < 





h 

The conditions at oo have the effect that (cf. (10.13.41) et seq.): 

(10.13.70) (p(x) = Be ikx + Re~ ikx , x > 0, 

(10.13.71) <p(x) = Te ikx , x < — I, 

with k = a/\/gh> All of this is the same as for the previous cases. We 
turn now to the conditions which result from the assumption that the 
floating body is a beam. 

The differential equation governing small transverse oscillations 
of a beam is 

(10.13.72) EIr] xxxx +mr] tt = p, 

in which E is the modulus of elasticity, / the moment of inertia of a 
cross section of unit breadth (or, perhaps better, EI is the bending 
stiffness factor), m the mass per unit area, and p is the pressure. We 
ignore the weight of the beam and at the same time disregard the 
contribution of the hydrostatic pressure term in p corresponding to 
the equilibrium position of the beam — i.e. the pressure here is that due 
entirely to the dynamics of the situation. Thus 

(10.13.73) p= - Q t - ggrj 

= (— ioqcp — Qgv)e iat . 

Insertion of this relation in (10.13.72) and use of (10.13.69) leads at 
once to the differential equation for cp(x): 

d 6 w og — mo 2 d 2 w G 2 Q 

(10.13.74 —L + ^ 1 + _^ y =. o, - I <x <0, 

dx 6 EI dx 2 EIh r 



LONG WAVES IN SHALLOW WATER 441 

that is valid under the beam. The case of greatest importance for us — 
that of a floating beam used as a breakwater— leads obviously to the 
boundary conditions for the beam which correspond to free ends, i.e. 
to the conditions that the shear and bending moments should vanish 
at the ends of the beam. These conditions in turn mean that r\ xx and 
r) xxx should vanish at the ends of the beam, and from (10.13.67) and 
(10.13.69) we thus have for <p the boundary conditions 

(10.13.75) — ^ = — ? = at x = 0, x = — I. 
dx* dx 5 

The transition conditions (10.13.66) require that 99 and (p x be conti- 
nuous at x = 0, x = — I, and this, in view of (10.13.70) and (10.13. 
71), requires that 

f cp(0) = B + R, wJO) = ikB - ikR 
10.13.76 \ rK ' ^ rxK ' 

\(p(- I) = Te- ik \ <p x (-l) = ikTe~ ikl . 

We remark once more that the constant B is assumed to be real, but 
that R and T will in general be complex constants, and that the real 
parts of and r\ as given by (10.13.67) are to be taken at the end. 

In order to solve our problem we must solve the differential equa- 
tion (10.13.74) subject to the conditions (10.13.75) and (10.13.76). 
A count of the relations available to determine the solution should be 
made: The general solution of (10.13.74) contains six arbitrary con- 
stants, and we wish to determine the constants R and T (the am- 
plitudes of the reflected and transmitted waves) occurring in (10.13. 
76) once the constant B (the amplitude of the incoming wave) has been 
fixed. In all there are thus eight constants to be found, and we have in 
(10.13.75) and (10.13.76) eight relations to determine them. Once 
these constants have been determined, the reflection and transmission 
coefficients are known, and the deflection of the beam can be found 
from (10.13.69). The maximum bending stresses in the beam can then 
be calculated from the usual formula: s = Mc/I, with M = EIrj xx 
and c the distance from the neutral axis to the extreme outer fibres of 
the beam. 

In principle, therefore, the solution of the problem is straightfor- 
ward. However, the carrying out of the details in the case of the beam 
of finite length is very tedious, involving as it does a system of eight 
linear equations for eight unknowns with complex coefficients. In 
addition, one must determine the roots of a sixth degree algebraic 
equation in order to find the general solution of (10.13.74). These 



442 WATER WAVES 

roots are in general complex numbers and they involve the essential 
parameters of the mechanical system. Thus it is clear that a dis- 
cussion of the behavior of the system in general terms with respect to 
arbitrary values of the parameters of the system is not feasible, and 
one must turn rather to concrete cases in which most of the parameters 
have been given specific numerical values. The results of some calcula- 
tions of this kind, for a case proposed as a practical possibility, will be 
given a little later. 

The case of a semi-infinite beam — i.e. a beam extending from x = 
to x = — oo — is simpler to deal with in that the conditions in the 
second line of (10.13.76) fall away, and the conditions (10.13.75) at 
x = — oo can be replaced by the requirement that cp be bounded at 
x = — oo. The number of constants to be fixed then reduces to four 
instead of eight, but the determination of the deflection of the beam 
still remains a formidable problem; we shall consider this case as well 
as the case of a beam of finite length. 

We begin the program indicated with a discussion of the general 
solution of the differential equation (10.13.74). Since it is a linear 
differential equation with constant coefficients we proceed in the 
standard fashion by setting q> = e*, inserting in (10.13.74), to find 
for x the equation 

(10.13.77) x Q + ax 2 + b = 

with 

pg — ma 2 _ a 2 o 

10.13.78 a=^— , & = _£_. 

V ' EI Elh 

This is a cubic equation in x 2 = (3, which happens to be in the standard 
form to which the Cardan formula for the roots of a cubic applies 
directly. For the roots fa of this equation one has therefore 

(10.13.79) fa = su + e 2 v 

{ P3 = £2u + £V 

with u and v defined by 



lb lb 2 a\\\\ { b lb 2 a?\\\ 

io.ia.80) .-(.-+(_ + _), ■.-(-— {-+ s ) ) 



and e the following cube root of unity: 
(10.13.81) £= =JL±V? 



LONG WAVES IN SHALLOW WATER 443 

The constant a is positive, since cr, the frequency of the incoming 
wave, is so small in the cases of interest in practice that gg is much 
larger than ma 2 . The constant b is obviously positive. Consequently 
the root ^ is real and negative since \u\ < | v | and v is negative. 
Thus the roots x 1 = + /?J /2 , x 2 = — /?J /2 are pure imaginary. The 
quantities /? 2 and /? 3 are complex conjugates, and their square roots 
yield two pairs of complex conjugates 

*3 = + P\", **=- PI'*' *5 = + #'*. *«=- #"• 

For /? 2 an d /? 3 we have 

1 \/3 

(10.13.82) 2 = — - (u + v) + i — (u — v), 

'— — 

1 a/3 

(10.13.83) /5 3 = (u + a) — z — (w — i;). 

_ z 

Thus /? 2 and /? 3 both have positive real parts. We suppose the roots 
x 3 , # 4 , x 5 , x 6 to be numbered to that x 3 and x 5 are taken to have posi- 
tive real parts, while x^ and x 6 have negative real parts. The general 
solution of (10.13.74) thus is 

(10.13.84) <p(x) = a x e*& + a 2 e** x + a 3 e x * x + a^ x + a 5 e** x + a 6 e*6*. 

In the case of a beam covering the whole surface of the water, i.e. 
extending from — oo to +00, the condition that op be bounded at 
x = ± 00 would require that a 3 = a 4 = a 5 = a 6 = since the ex- 
ponentials in the corresponding terms have non-vanishing real parts. 
The remaining terms yield progressing waves traveling in opposite 
directions; their wave lengths are given by A = 2ji/\ x x \ = 2nj\ k 2 | 
and thus by 



(10.13.85) X = 2njV\u + v |, 

with u -f- v defined by (10.13.80). The wave length and frequency are 
thus connected by a rather complicated relation, and, unlike the case 
of waves in shallow water with no immersed bodies or constraints on 
the free surface, the wave length is not independent of the frequency 
and the wave phenomena are subject to dispersion. 

In the case of a beam extending from the origin to — 00 while the 
water surface is free for x > 0, the boundedness conditions for cp at 
— co requires that we take a 4 = a 6 = since x^ and x 6 have negative 
real parts and consequently e x * x and e*« x would yield exponentially 
unbounded contributions to 99 at x = — 00. We know that x x and x 2 



444 



WATER WAVES 



are pure imaginary with opposite signs, with x 2 , say, negative imagin- 
ary. Since no progressing wave is assumed to come from the left, we 



must then take a 9 = 0. Thus the term 



yields the transmitted 



wave and the terms involving a 3 and a b yield disturbances which die 
out exponentially at go. The conditions (10.13.70) and (10.13.71) at 
x = now yield the following four linear equations: 



(10.13.86 



x\a x -f x*a 3 -f- x\a h = 
%\a x + n\a z + x\a b = 

«i + a z + H = B + R 

7i x a x + x s a 3 + x 5 a 5 = ik(B - R) 



for the constants a v a 3 , a 5 , R. For the amplitude R of the reflected 
wave one finds 



(10.13.87) 



1 



1 



1 






^5 


ik 


A 





A 





- i 


+ 1 


*5 


ik 


^ 





< 






Even in this relatively simple case of the semi-infinite beam the re- 
flection coefficient is so complicated a function of the parameters 
(even though it is algebraic in them) that it seems not worthwhile to 
write it down explicitly. The results of numerical calculations based 
on (10.13.87) will be given shortly. 

In the case of the beam of finite length extending from x = — I 
to x = the eight conditions given by (10.13.75) and (10.13.76) must 
be satisfied by the solution (10.13.84) of the differential equation 
(10.13.74), and these conditions serve to determine the six constants 
of integration and the amplitudes R and T of the reflected and trans- 
mitted waves. The problem thus posed is quite straightforward but 
extremely tedious as it involves solving eight linear equations for 
eight complex constants. For details reference is again made to the 
work of Wells [F.5]. 

This case of a floating beam was suggested to the author by J. H. 



LONG WAVES IN SHALLOW WATER 445 

Carr of the Hydraulics Structures Laboratory at the California Insti- 
tute of Technology as one having practical possibilities; at his sug- 
gestion calculations in specific numerical cases were carried out 
in order to determine the effectiveness of such a breakwater. The 
reason for considering such a structure for a breakwater as a 
means of creating relatively calm water between it and the shore 
is the following: a structure which floats on the surface without sink- 
ing far into the water need not be subjected to large horizontal forces 
and hence would not necessarily require a massive anchorage. How- 
ever, in order to be effective as a reflector of waves such a floating 
structure would probably have to be built with a fairly large dimen- 
sion in the direction of travel of the incoming waves. As a consequence 
of the length of the structure, it would be bent like a beam under the 
action of the waves and hence could not in general be treated with 
accuracy as a rigid body in determining its effectiveness as a barrier. 
This brings with it the possibility that the structure might be bent 
so much that the stresses set up would be a limiting feature in the 
design. The specifications (as suggested by Carr) for a beam having 
a width of one foot (parallel to the wave crest, that is) were: 

Weight: 85 pounds j ft 2 

Moment of inertia (of area of cross-section): 0.2 // 4 

Modulus of elasticity: 437 X 10 7 pounds j ft 2 . 

The depth of the water is taken as 40 feet. Simple harmonic progres- 
sing waves having periods of 8 and of 15 sees, were to be considered, 
and these correspond to wave lengths of 287 and 539 feet, and to cir- 
cular frequencies o of 785 x 10~ 3 and 418 x 10~ 3 cycles per second, 
respectively. The problem is to determine the reflecting power of the 
beam under these circumstances when the length of the beam is varied. 
In other words, we assume a wave train to come from the right hand 
side of the beam and that it is partly transmitted under the beam to 
the left hand side and partly reflected back to the right hand side. 
The ratio R/B of the amplitude R of the reflected wave and the am- 
plitude B of the incoming wave is a measure of the effectiveness of 
the beam as a breakwater. 

Before discussing the case of beams of finite length it is interesting 
and worthwhile to consider semi-infinite beams first. Since the calcu- 
lations are easier than for beams of finite length it was found possible 
to consider a larger range of values of the parameters than was given 



446 WATER WAVES 

above. The results are summarized in the following tables (taken 
from [F.5]): 

TABLE A 



I (ft) 


\secj 


W (pounds) 


lit*) 


/ pounds \ 

E ( a* ) 


h(jt) 


R/B 


539 


0.418 


85 


0.20 


437 X 10 7 


40 


0.14 


287 


0.785 


85 


0.20 


437 X 10 7 


40 


0.19 


225 


1.0 


85 


0.20 


437 X 10 7 


40 


0.23 


150 


1.5 


85 


0.20 


437 X 10 7 


40 


0.32 


113 


2.0 


85 


0.20 


437 X 10 7 


40 


0.43 



In Table A the beam design data are as given above. At the two speci- 
fied circular frequencies of 0.418 and 0.785 one sees that the floating 
beam is quite ineffective as a breakwater since the reflected wave has 
an amplitude of less than 1/5 of the amplitude of the incoming wave, 
even for the higher frequency (and hence shorter wave length), which 
means that less than 4 % of the incoming energy is reflected back. At 
higher frequencies, and hence smaller wave lengths, the breakwater 
is more effective, as one would expect. However the approximate 
theory used to calculate the reflection coefficient R/B can be expected 
to be accurate only if the ratio X/h of wave length to depth is suffi- 
ciently large, and even for the case X = 287 ft. (a — .785) the re- 
flection coefficient of value 0.19 may be in error by perhaps 10 % or 
more since X/h is only about 7, and the errors for the shorter wave 
lengths would be greater. Calculations for still other values of the 
parameters are shown in Table B. The only change as compared with 









TABLE B 






I 


a 


W 


I E 


h 


R/B 


539 

287 


0.418 
0.785 


384 
384 


0.20 437 X 10 7 
0.20 437 X 10 7 


40 
40 


0.51 
0.75 



the first two rows of Table A is that the weight per foot of the beam 
has been increased by a factor of more than 4 from a value of 85 
pounds/ ft 2 to a value of 384 pounds/ ft 2 . The result is a decided increase 
in the effectiveness of the breakwater, especially at the shorter wave 
length, since more than half (i.e. (.75) 2 ) of the incoming energy would 
be reflected back. However, this beneficial effect is coupled with a 
decided disadvantage, since quadrupling the weight of the beam 



LONG WAVES IN SHALLOW WATER 447 

would cause it to sink deeper in the water in like proportion and hence 
might make heavy anchorages necessary. Table C is the same as the 









TABLE 


c 






X 


G 


W 


/ 


E 


h 


R/B 


539 


.418 


85 


2.0 


437 X 10 7 


40 


.26 


287 


.785 


85 


2.0 

00 


437 X 10 7 


40 


.32 
1 



first two rows of Table A except that the bending stiffness has been 
increased by a factor of 10 by increasing the moment of inertia of the 
beam cross-section from 0.2 ft* to 2.0 ft*. Such an increase in stiffness 
results in a noticeable increase in the effectiveness of the breakwater, 
but by far not as great an increase as is achieved by multiplying the 
weight by a factor of four. If the stiffness were to be made infinite 
(i.e. if the beam were made rigid) the reflection coefficient could be 
made unity, and no wave motion would be transmitted. This is 
evidently true for a semi-infinite beam, but would not be true for a 
rigid body of finite length. 







TABLE D 






X 


a 


W I E 


h 


RIB 


539 

287 


.418 

.785 


85 
85 


40 
40 


.001 
.007 



In Table D the difference as compared with Table A is that the 
beam stiffness is taken to be zero. This means that the surface of the 
water is assumed to be covered by a distribution of inert particles 
weighing 85 pounds per foot. (Such cases have been studied by Gold- 
stein and Keller [G.l].) As we observe, there is practically no reflec- 
tion and this is perhaps not surprising since the mass distribution per 
unit length has such a value that the beam sinks down into the water 
only slightly. 

One might perhaps summarize the above results as follows: A very 
long beam can be effective as a floating breakwater if it is stiff enough. 
However, a reasonable value for the stiffness (the value 0.2 given 
above) leads to an ineffective breakwater unless the weight of the 
beam per square foot is a fairly large multiple (say 8 or 10) of the 
weight of water. 

In practice it seems unlikely that beams long enough to be considered 



448 WATER WAVES 

semi-infinite would be practicable as breakwaters. (The term "long 
enough" might be interpreted to mean a sufficiently large multiple of 
the wave length, but since the wave lengths are of the order of 200 
feet or more the correctness of this statement seems obvious.) It there- 
fore is necessary to investigate the effectiveness of beams of finite 
length. Such an investigation requires extremely tedious calculations 
— so much so that only a certain number of numerical cases have been 
treated. These are summarized in the following tables. 



g = .785, 


X = 287 


a = .418, 


X = 539 


l(ft) 


R\B 


I (ft) 
145.9 


RIB 


17.5 





.17 


49.2 


.93 


196.9 


.53 


72.9 





291.8 


.13 


98.5 


.75 


443.0 


.90 


145.9 


.10 


583.6 


.74 


196.9 





656.2 


.62 


291.8 


.33 


874.9 


.07 


450.4 


.32 


875.4 


.08 


583.6 


.12 


948.3 


.54 


656.3 


.13 
.32 


oo 


.14 


875.4 






oo 


.19 







In these tables the parameters have values the same as in the first 
two rows of Table A, except that now lengths other than infinite 
length are considered. The most noticeable feature of the results given 
in the tables is their irregularity and the fact that at certain lengths 
—even certain rather short lengths — the beam proposed by Carr 
seems to be quite effective. For example, when the wave length is 
287 ft. a beam less than 50 ft. long reflects more than 80 % of the 
incoming energy. A beam of length 443 ft. is also equally effective at 
the longer wave length of 539 ft.* 

* It might not be amiss to consider the physical reason why it is possible that 
a beam of finite length could be more effective as a breakwater than a beam of 
infinite length. Such a phenomenon comes about, of course, through multiple 
reflections that take place at the ends of the beam. Apparently the phases some- 
times arrange themselves in the course of these complicated interactions in such 
a way as to yield a small amplitude for the transmitted wave. That such a process 
might well be sensitive to small changes in the parameters, as is noted in the 
discussion, cannot be wondered at. 



LONG WAVES IN SHALLOW WATER 449 

However, the maximum effectiveness of any such breakwater 
occurs for a specific wave length within a certain range of wave 
lengths; thus the reflection of a given percentage of the incoming wave 
energy would involve changing the length (or some other parameter) of 
the structure in accordance with changes in the wave length of the 
incoming waves. Also, the reflection coefficient seems to be rather 
sensitive to changes in the parameters, particularly for the shorter 
structures (a relatively slight change in length from an optimum value, 
or a slight change in frequency, leads to a sharp decrease in the re- 
flection coefficient). It is also probable — as was indicated earlier on the 
basis of calculations by Wells [W.10] —that the shallow water approx- 
imation used here as a basis for the theory is not sufficiently accurate 
for a floating beam whose length is too much less than the wave length. 
Nevertheless, it does seem possible to design floating breakwaters of 
reasonable length which would be effective at a given wave length. 
Perhaps it is not too far-fetched to imagine that sections could be 
added to or taken away from the breakwater in accordance with 
changing conditions. 

Another consequence of the theory— which is also obvious on 
general grounds— is that there is always the chance of creating a 
large standing wave between the shore and the breakwater because of 
reflection from the shore, unless the waves break at the shore; this 
effect is perhaps not important if the main interest is in breakwaters 
off beaches of not too large slope, since breaking at the shore line then 
always occurs. (The theory developed here could be extended to cases 
in which the shore reflects all of the incoming energy, it might be 
noted.) In principle, the calculation of the deflection curve of the 
structure, and hence also of the bending stresses in it, as given by the 
theory is straightforward, but it is very tedious; consequently only 
the reflection coefficients have been calculated. 



CHAPTER 11 



Mathematical Hydraulics 



In this chapter the problems to be treated are, from the mathema- 
tical point of view, much like the problems of the preceding chapter, 
but the emphasis is on problems of rather concrete practical signifi- 
cance. Aside from this, the essential difference is that external forces 
other than gravity, such as friction, for example, play a major role in 
the phenomena. Problems of various types concerning flows and 
wave motions in open channels form the contents of the chapter. The 
basic differential equations suitable for dealing with such flows under 
rather general circumstances are first derived. This is followed by a 
study of steady motions in uniform channels, and of progressing waves 
of uniform shape, including roll waves in inclined channels. Flood 
waves in rivers are next taken up, including a discussion of numerical 
methods appropriate in such cases; the results of such calculations 
for a flood wave in a simplified model of the Ohio River and for a 
model of its junction with the Mississippi are given. This discussion 
follows rather closely the two reports made to the Corps of Engineers 
of the U.S. Army by Stoker [S.23] and by Isaacson, Stoker, and 
Troesch [1.4]. These methods of dealing with flood waves have been 
applied, with good results, to a 400-mile stretch of the Ohio as it 
actually is for the case of the big flood of 1945, and also to a flood 
through the junction of the Ohio and the Mississippi; these results 
will be discussed toward the end of this chapter. 

There is an extensive literature devoted to the subject of flow in 
open channels. We mention here only a few items more or less directly 
connected with the material of this chapter: the famous Essai of 
Boussinesq [B.17], the books of Bakhmeteff [B.3] and Rouse [R.10, 
11] (in particular, the article by Gilcrest in [R.ll]), the Enzyklopadie 
article of Forchheimer [F.6] and the booklet by Thomas [T.2]. 

451 



452 



WATER WAVES 



11.1. Differential equations of flow in open channels 

It has already been stated that the basic mathematical theory to 
be used in this chapter does not differ essentially from the theory 
derived in the preceding chapter. However, there are additional 
complications due to the existence of significant forces beside gravity, 
and we wish to permit the occurrence of variable cross-sections in the 
channels. Consequently the theory is derived here again, and a some- 
what different notation from that used in previous chapters is em- 
ployed both for the sake of convenience and also to conform somewhat 
with notations used in the engineering literature. 

The theory is one-dimensional, i.e. the actual flow in the channel is 
assumed to be well approximated by a flow with uniform velocity over 
each cross-section, and the free surface is taken to be a level line in 
each cross-section. The channel is assumed also to be straight enough 
so that its course can be thought of as developed into a straight line 
without causing serious errors in the flow. The flow velocity is denoted 
by v, the depth of the stream (commonly called the stage in the 
engineering literature) by y, and these quantities are functions of the 










, 










7>>^ 






h 


i 


r^-—^^^ 










y 


Z 




— *■ 


I Ax 

i 


>— 






7>T»> 


r 





Fig. 11.1.1. River cross-section and profile 

distance x down the stream and of the time t (cf. Fig. 11.1.1). The 
vertical coordinates of the bottom and of the free surface of the stream, 
as measured from the horizontal axis x, are denoted by z(x) and 



MATHEMATICAL HYDRAULICS 453 

h(x, t), with z positive downward, h positive upward; thus y = h -f z. 
The slope of the bed is therefore counted positive in the positive 
^-direction, i.e. downward. The breadth of the free surface at any 
section of the stream is denoted by B. 

The differential equations governing the flow are expressions of the 
laws of conservation of mass and momentum. In deriving them the 
following assumptions, in addition to those mentioned above, are 
made * : 1 ) the pressure in the water obeys the hydrostatic pressure 
law, 2 ) the slope of the bed of the river is small, 3 ) the effects of friction 
and turbulence can be accounted for through the introduction of a 
resistance force depending on the square of the velocity v and also, in 
a certain way to be specified, on the depth y. 

We first derive the equation of continuity from the fact that the 
mass gAAx included in a layer of water of density g, thickness Ax, 
and cross-section area A, changes in its flow along the stream only 
through a possible inflow along the banks of the stream, say at the 
rate gq per unit length along the river. The total flow out of the ele- 
ment of volume A Ax is given by the net contributions g(Av) x Ax from 
the flow through the vertical faces plus the contribution gBh t Ax due 
to the rise of the free surface, with B the width of the channel; since 
Bh t represents the area change A t it follows that the sum [(Av) x + 
A t ]Ax equals the volume influx qAx over the sides of the channel, 
with q the influx per unit length of channel. The subscripts x and t 
refer, of course, to partial derivatives with respect to these variables. 
The equation of continuity therefore has the form 

(11.1.1) (Av) 9 +A t = q. 

It should be observed that A = A(y(x, t), x) is in the nature of 
things a given function of y and x, although y(x, t) is an unknown 
function to be determined; in addition, q = q(x, t) depends in general 
on both x and < in a way that is supposed given. In the important 
special case of a rectangular channel of constant breadth B, so that 
A = By, the equation of continuity takes the form 

(11.1.2) v x y + vy x +y t = q/B. 

The equation of motion is next derived for the same slice of mass 
m = gAAx by equating the rate of change of momentum d(mv)jdt 

* These assumptions are not the minimum number necessary: for example, 
assumption 1 ) has as a consequence the independence of the velocity on the vertical 
coordinate if that were true at any one instant (cf. the remarks on this point 
in Ch. 2 and Ch. 10). 



454 WATER WAVES 

to the net force on the element. We write the equation of motion for 
the horizontal direction: 

(11.1.3) g — (AvAx) = HAx — F f Ax cos <p+ggAAx sin cp. 
dt 

In this equation H represents the unbalanced horizontal pressure 
force at the surface of the element. The angle cp is the slope angle of the 
bed of the channel, reckoned positive downward. The quantity F f re- 
presents the friction force along the sides and bottom of the channel, 
and the term qgAAx sin cp represents the effect of gravity in accelerat- 
ing the slice down-hill as manifested through the normal reaction of 
the stream bed. Since cp was assumed small we may replace sin cp by 
the slope S = dz/dx and cos cp by 1. In the frictional resistance term 
we set 

This is an empirical formula called Manning's formula. The resistance 
is thus proportional to the square of the velocity and is opposite to its 
direction; in addition, the friction is inversely proportional to the 
4/3-power of the hydraulic radius R, defined as the ratio of the cross- 
section area A to the wetted perimeter (thus R = By/(B -\- 2y) for a 
rectangular channel and R = y for a very wide rectangular channel), 
and inversely proportional to y, a roughness coefficient. 

We calculate next the momentum change Qd(AvAx)/dt. In doing so, 
we observe that the symbol d/dt must be interpreted as the particle 
derivative (cf. Chapter 1.1 and equation (1.1.3)) d/dt -\- vd/dx since 
Newton's law must be applied in following a given mass particle along 
its path x = «(£). However, the law of continuity (11.1.1) derived above 
is clearly equivalent to writing d(AAx)/dt = qAx, with d/dt again in- 
terpreted as the particle derivative. Since 

d , . A , d , A . „ „ . dv 

— (AvAx)=v — (AAx) J r AAx — 
dt dt dt 

it follows that 

— (AvAx)=AAx(vv x J r v t ) J r qvAx. 
dt 

Finally, the net contribution HAx of the pressure forces over the 
surface of the slice is calculated as follows : The total pressure over a 

vertical face of the slab is given by | Qg[y(x, t) — £]b(x, £) d£ from 

the hydrostatic pressure law (cf. Fig. 11.1.1); while the component 



MATHEMATICAL HYDRAULICS 455 

in the ^-direction of the total pressure over the part of the slice in 
contact with the banks of the river is given by 

( I * QglV ~ £]bxi x > £) d£ Ax, we have for HAx the following equation: 
(11.1.5) HAx = - |- | r qg[y(x, t)-^]b(x, £)d£ J Ax 



j" 

Jo 



QgyxHx, £)d£=-QgAy x . 



In this calculation the integrals involving b x cancel out, and we have 
used the fact that y x is independent of £. 

Adding all of the various contributions we have 

(H.1.6) v t +vo; p + i v=Sg-S f g-gy x 

upon defining what is called the friction slope ^ by the formula 

(11.1.7) S t = ±-F t% 

with F f defined by (11.1.4). It should perhaps be mentioned that the 
term qvjA on the left hand side of (11.1.6) arises because of the 
tacit assumption that flows enter the main stream from tributaries 
or by flow over the banks at zero velocity in the direction of the main 
stream; if such flows were assumed to enter with the velocity of the 
main stream, the term would not be present — it is, in any case, a 
term which is quite small. If we introduce A = A(y(x, t), x) in 
(11.1.1) the result is 

(11.1.8) A y y x v + A x v + Av x + A y y t = q. 

The two differential equations (11.1.6) and (11.1.8), which serve 
to determine the two unknown functions, the depth y(x, t) and the 
velocity v(x, t), are the basic equations for the study of flood waves in 
rivers and flows in open channels generally. For any given river or 
channel it is thus necessary to have data available for determining 
the cross-section area A and the quantities y and R in the resistance 
term F f as functions of x and y, and of the slope S of its bed as a 
function of x in order to have the coefficients in the differential equa- 
tions (11.1.6) and (11.1.8) defined. Three of these quantities are 
purely geometrical in character and could in principle be determined 



456 WATER WAVES 

from an accurate contour map of the river valley, but the determina- 
tion of the roughness coefficient y of course requires measurements of 
actual flows for its determination. 

11.2. Steady flows. A junction problem 

We define a steady flow in the usual fashion to be one for which the 
velocity v and depth y are independent of the time, that is, v t =y t =0. 
In this section channels of constant rectangular cross-section and 
constant slope will be considered for the most part. It follows from the 
equation of continuity (cf. (11.1.2)): 

V% + vy x + yv x = 0, 
that for steady flow 

(11.2.1) {vy) x = whence vy = D (Da constant), 

when no flow into the channel from its sides occurs (i.e. q = in 
(11.1.2)). Similarly, the equation of motion (cf. (11.1.6)) 

v t + vv x + gy x + g(S f - S) = 
yields 

(11.2.2) vv x + gy x + g(S f -S) = 0. 

It follows from equation (11.2.1) that 

D A D 

v = - and v x = - — y x , 

y y 2 

so that equation (11.2.2) becomes 

/ # 2 \ / D 2 \ 

(n.2.3) [g- —)y x +g — r— ^tt^ -S\=o. 

1 y ' w 



H + nr 



Here the hydraulic radius is given by K = yj(\ -\- 2y/B) because the 
channel is assumed to be rectangular in cross-section. 

For a channel with given physical parameters such as cross-section, 
resistance coefficient, etc. the steady flows would provide what are 
called backwater curves. In general, one could in principle always 
find steady solutions y = y(oc) and v = v(x) for a non-uniform chan- 
nel. The explicit determination of the stage y and discharge rate BD 
as functions of x would be possible by numerical integration of the 
pair of first order ordinary differential equations arising from (11.1.6) 
and (11.1.8) when time derivatives are assumed to vanish. 



MATHEMATICAL HYDRAULICS 457 

We note that equation (11.2.3) has the simple solution y = constant 
for y satisfying 

This means that we can find a flow of uniform depth and velocity 
having a constant discharge rate BD (B is, as in the preceding section, 
the width of the channel). Conversely, by fixing the depth y we can 
find the discharge from (11.2.4) appropriate to the corresponding 
uniform flow. Physically this means that the flow velocity is chosen 
so that the resistance due to turbulence and friction and the effect of 
gravity down the slope of the stream just balance each other. We re- 
mark that if (11.2.4) is satisfied at any point where the coefficient 
g — D 2 /y 3 of y x in (11.2.3) does not vanish, then y = constant is the 
only solution of (11.2.3) because of the fact that the solution is then 
uniquely determined by giving the value of y at any point x. We note 
that g — D 2 /y 3 = corresponds to v = Vg2/> l - e - *° a fl° w a t critical 
speed (a term to be discussed in the next section), since D = vy. 
Furthermore, the differential equation (11.2.3) can be integrated to 
yield a; as a function of y: 

when x = for y = y . 

We proceed to make use of (11.2.5) in order to study a problem 
involving a steady flow at the junction of two rivers each having a 
rectangular channel. Later on, the same problem will be treated but 
for an unsteady motion resulting from a flood wave traveling down 
one of the branches, and such that the steady flow to be treated here 
is expected to result as a limit state after a long time. The numerical 
data are chosen here for the problem in such a way as to correspond 
roughly with the actual data for the junction of the Ohio River with 
the Mississippi River. Thus the Ohio is assumed to have a rectangular 
channel 1000 feet in width and a constant slope of .5 feet /mile. In 
Manning's formula for the resistance the constant y is assumed given 
by 7 — (1.49/w) 2 in terms of Manning's roughness coefficient n, and 
n is given the value 0.03. The upstream branch of the Mississippi was 
taken the same in all respects as the Ohio, but the downstream branch 
is assumed to have twice the breadth, i.e. 2000 feet, and its slope to 



458 WATER WAVES 

have a slightly smaller value, i.e. 0.49 feet/mile instead of 0.5 feet/mile. 
With these values of the parameters, a flow having the same uniform 
depth of 20 feet in all three branches is possible — the choice of the 




Lower 
Mississippi 



Fig. 11.2.1. Junction of Ohio and Mississippi Rivers 

value 0.49 feet/mile for the slope of the downstream branch of the 
Mississippi River was in fact made in order to ensure this. Later on 
we intend to calculate the progress of a flood which originates at a 
moment when the flow is such a uniform flow of depth 20 feet. The 
flood wave will be supposed to initiate at a point 50 miles up the Ohio 
from the junction and to be such that the Ohio rises rapidly at that 
point from the initial depth of 20 feet to a depth of 40 feet in 4 hours. 
A wave then moves down the Ohio to the junction and creates waves 
which travel both upstream and downstream in the Mississippi as well 
as a reflected wave which travels back up the Ohio. After a long time 
we would expect a steady state to develop in which the depth at the 
point 50 miles up the Ohio is 40 feet, while the depth far upstream in 
the Mississippi would be the original value, i.e. 20 feet (since we would 
not expect a retardation of the flow far upstream because of an inflow at 
the junction). Downstream in the Mississippi we expect a change in the 
flow extending to infinity. It is the steady flow with these latter charac- 
teristics that we wish to calculate in the present section. ( See Fig. 11.2.1) 
We remark first of all that the stream velocities in all of the three 
branches will always be subcritical — in fact, they are of the order of 
a few miles per hour while the critical velocities <\/gy are of the order 
of 15 to 25 miles per hour. It follows that the quantity g — D 2 /y 3 in the 
integrand of the basic formula (11.2.5) for the river profiles (i.e. the 
curve of the free surface) is always positive. The integrand I(y) in 



MATHEMATICAL HYDRAULICS 



459 



that formula has the general form indicated by Fig. 11.2.2 in the case 
of flows at subcritical velocities. The vertical asymptote corresponds 
to the value of y for which a steady flow of constant depth exists 




Fig. 11.2.2. The integrand in the wave profile formula 



(cf. (11.2.4)), since the square bracket (the denominator in the inte- 
grand) vanishes for this value. It follows that x can become positive 
infinite for finite values of y only if y takes on somewhere this value; 
but in that case we have seen that the whole flow is then one with 
constant depth everywhere. Consequently the downstream side of the 
Mississippi carries a flow of constant speed and depth, though the 
values of these quantities are not known in advance. However, in the 
upstream branch of the Mississippi the flow need not be constant, and 
of course we do not expect it to be constant in the Ohio: in these 
branches x must be taken to be decreasing on going upstream and 
consequently the negative portion of I(y) indicated in Fig. 11.2.2 
comes into use since we may, and do, set x = at the junction. 
For the sake of convenience we use subscripts 1, 2, and 3 to refer to 
all quantities in the Ohio, the upstream branch of the Mississippi, and 
the downstream branch of the Mississippi respectively. The conditions 
to be satisfied at the junction are chosen to be 
(H-2.6) yi = y 2 = y 3 = y. 

(11.2.7) D 1 +D 2 = 2D 3 



for x = 0. 



460 WATER WAVES 

The first condition simply requires the water level to have the same 
value y i (which is, however, not known in advance) in all three bran- 
ches, while the second states, upon taking account of the first condition, 
that the combined discharge of the two tributaries makes up the total 
discharge in the main stream. The quantity D 2 , the discharge in the 
upper Mississippi, is known since the flow far upstream in this branch 
is supposed known— i.e. it is a uniform flow of depth 20 feet. 

By using (11.2.7) in (11.2.4) as applied to the lower branch of the 
Mississippi (in which the flow is known to be constant) we have 



Next, we write equation (11.2.5) for the 50-mile stretch of the Ohio 
which ends at the point where the depth in that branch was prescribed 
to be 40 feet (and which was the point of initiation of a flood wave); 
the result is 

(11.2.9) 50= j I(y,D v B 1 )dy 






in which it is indicated that D and B (as well as all other parameters) 
are to be evaluated for the Ohio; the quantity y has the value 40/5280 
in miles. Equations (11.2.8) and (11.2.9) are two equations containing 
y j and Z) 1 as unknowns, since D 2 is known. They were solved by an 
iterative process, i.e. by taking for D x an estimated value, determining 
a value for y 6 from (11.2.9), reinserting this value in (11.2.8) to deter- 
mine a new value for D l9 etc. The results obtained by such a calcula- 
tion are as follows: 

yi = V2 = Vz = Vi = 81-2 feet 
v 1 = 4.83 miles/hour, v 2 = 1.53 miles/hour, v 3 = 3.18 miles/hour. 

The profiles of the river surface can now be computed from (11.2.5); 
the results are given in Fig. 11.2.3. 

The solution of the mathematical problem has the features we 
would expect in the physical problem. The flow velocity and stage are 
increased at the junction, even quite noticeably, by the influx from 
the Ohio. Upstream in the Mississippi the stage decreases rather 
rapidly on going away from the junction, and very little backwater 
effect is noticeable at distances greater than 50 miles from the junc- 
tion. This illustrates a fact of general importance, i.e. that backwater 



MATHEMATICAL HYDRAULICS 



461 



effects in long rivers arising from even fairly large discharges of tri- 
butaries into the main stream do not persist very far upstream, 
but such an influx has an influence on the flow far downstream. 
For unsteady motions this general observation also holds, and is in 
fact one of the basic assumptions used by hydraulics engineers in 



feet 



Ohio 




Upstream 
Mississippi 



40 



Downstream 
Mississippi 



20 



■*- x 
miles 



-50 Junction 50 100 

Fig. 11.2.3. Steady flow profile in a model of the Ohio and Mississippi Rivers 

computing the passage of flood waves down rivers (a process called 
flood routing by them). Later on, in sec. 6 of this chapter, we shall 
deal with the unsteady motion described above in our model of the 
Ohio-Mississippi system, and we will see that the unsteady motion 
approaches the steady motion found here as the time increases. 

11.3. Progressing waves of fixed shape. Roll waves 

In addition to the uniform steady flows treated above there also 
exist a variety of possible flows in uniform channels in the form of 
progressing waves moving downstream at constant speed without 
change in shape. Such waves are expressed mathematically by depths 
y(x, t) and velocities v(x, t) in the form 
(11.3.1) y(x,t) = y(x — Ut), v(x, i) = v(x — Ut), U = const. 



462 WATER WAVES 

The constant U is of course the propagation speed of the wave as 
viewed from a fixed coordinate system; if viewed from a coordinate 
system moving downstream with constant velocity U the wave profile 
would appear fixed, and the flow would appear to be a steady flow 
relative to the moving system. It is convenient to introduce the va- 
riable £ by setting 

(11.3.2) f = X .-Ut 

so that y and v are functions of £ only. In this case the equations of 
continuity and motion given by (11.1.6) and (11.1.8) become, for a 
rectangular channel of fixed breadth and slope: 

(11.3.3) I (v-U)y c +yv ( = 0, 

with S the slope of the channel and S f defined, as before, by 



(11.3.4) S f = 



v\v.\ 



('+!))• 



y\y\ 



The first equation of (11.3.3) can be integrated to yield 

(11.3.5) (v - U)y = D = const. 

as one readily verifies, and the second equation then takes the form 

(11.3.6) (g-_ \y,+g(S f -S)=0. 



(g-p)vc+glS,- 



The first order differential equation (11.3.6) has a great variety of 
solutions, which have been studied extensively, for example by Tho- 
mas [T.l], but most of them are not very interesting from the physical 
point of view. However, one type of solution of (11.3.6) is particularly 
interesting from the point of view of the applications, and we there- 
fore proceed to discuss it briefly. The solution in question furnishes 
the so-called monoclinal rising flood wave in a uniform channel (see 
the article by Gilcrest in the book of Rouse [R.ll, p. 644]). This, as 
its name suggests, is a progressing wave the profile of which tends to 
different constant values (and the flow velocity v also to different con- 
stant values) downstream and upstream, with the lower depth down- 
stream, connected by a steadily falling portion, as indicated schema- 
tically in Fig. 11.3.1. In this wave the propagation speed U is larger 



MATHEMATICAL HYDRAULICS 



463 



than the flow velocity v. It is always a possible type of solution of 
(11 .3.6) if the speed of propagation of the wave relative to the flow is 



<cV> 




Fig. 11.3.1. Monoclinal rising flood wave 

subcritical, i.e. if (U—v) 2 is less than gy, in which case the coefficient 
of the first derivative term in (11.3.6) is seen to be positive. This can 
be verified along the following lines. The differential equation can be 
solved explicitly for J as a function of y: 



(11.3.7; 



c 



J y* 



i{y)dy> 



with the integrand I(y) defined in the obvious manner; here y* is the 
value of y corresponding to f = 0. The function I(y) has the general 



Ky)' 
1 




_L 


s 


/ 1 




^^ y ° 


r | y . • y 



Fig. 11.3.2. The integrand I(y) for a monoclinal wave 



464 WATER WAVES 

form shown in Fig. 11.3.2 if the propagation speed U and the constant 
D in (11.3.5) are chosen properly. The main point is that the curve has 
two vertical asymptotes at y = y and y = y x between which I(y) 
is negative. By choosing y* between y and y 1 we can hope that 
£ -> + oo as y -> y , while f -> — oo as y -> y x : all that is necessary 
is that I(y) becomes infinite at y and y 1 of sufficiently high order. 
This is, in fact, the case; we can select values of D and U in such a 
way that I(y) becomes infinite at y and y x through setting the quan- 
tity S f — S in (11.3.6) equal to zero, i.e. by choosing D and U such that 



(11.3.8 



(Uy +D)< 



(Uy.+D] 




For given positive values of y and y x these are a pair of linear equa- 
tions (after taking a square root) which determine U and D uniquely. 
An elementary discussion of the possible solutions of these equations 
shows that U must be positive and D negative, and this means, as we 
see from (11.3.5), that U is larger than v, i.e. the propagation speed of 
the wave is greater than the flow speed. 

By taking the numerical data for the model of the Ohio given in the 
preceding section and assuming the depth far upstream to be 40 feet, 
far downstream 20 feet, it was found that the corresponding mono- 
clinal flood wave in the Ohio would propagate with a speed of 5 
miles/hour. The shape of the wave will be given later in sec. 6 of this 
chapter, where it will be compared with an unsteady wave obtained by 
gradually raising the level in the Ohio at one point from 20 feet to 
40 feet, then holding the level fixed there at the latter value, and cal- 
culating the downstream motion which results. We shall see that the 
motion tends to the monoclinal flood wave obtained in the manner 
just now described. Thus the unsteady wave tends to move eventually 
at a speed of about 5 miles/hour, while on the other hand, as we know 
from Chapter 10 (and will discuss again later on in this chapter), the 
propagation speed of small disturbances relative to the stream is y/gy 
and hence is considerably larger in the present case, i.e. of the order 
of 15 to 25 miles/hour. This important and interesting point will be 
discussed in sec. 6 below. 



MATHEMATICAL HYDRAULICS 



465 




Fig. 11.3.3. Roll waves, looking down stream (The Grunnbach, Switzerland) 






466 



WATER WAVES 



We turn next to another type of progressing waves in a uniform 
channel which can be described with the aid of the differential equa- 
tion (11.3.6), i.e. the type of wave called a roll wave. A famous exam- 
ple of such waves is shown in Fig. 11.3.3, which is a photograph taken 
from a book of Cornish [C.7], and printed here by the courtesy of the 
Cambridge University Press. As one sees, these waves consist of a 
series of bores (cf. Chapter 10.7) separated by stretches of smooth 
flow. The sketch of Fig. 11.3.4 indicates this more specifically. Such 




77777777777777777777777777777777777777777777777777777777777777777777777777777777777; 
Fig. 11.3.4. Roll waves 



waves frequently occur in sufficiently steep channels as, for example, 
spill-ways in dams or in open channels such as that of Fig. 11.3.3. 
Roll-waves sometimes manifest themselves in quite unwanted places, 
as for example in the Los Angeles River in California. The run-off from 
the steep drainage area of this river is carried through the city of Los 
Angeles by a concrete spill- way; in the brief rainy season a large 
amount of water is carried off at high velocity. It sometimes happens 
that roll waves occur with amplitudes high enough to cause spilling 
over the banks, though a uniform flow carrying the same total amount 
of water would be confined to the banks. The phenomenon of roll 
waves thus has some interest from a practical as well as from a theore- 
tical point of view; we proceed to give a brief treatment of it in the 
remainder of this section following the paper of Dressier [D.12]. In 
doing so, we follow Dressier in taking what is called the Chezy for- 
mula for the resistance rather than Manning's formula, as has been 
done up to now. The Chezy formula gives the quantity S f the following 
definition: 



(11.3.9) 



S f = 



v\v\ 



gR 



in which r 2 is a "roughness coefficient" and R is, as before, the hy- 
draulic radius. For a verv broad rectangular channel, the onlv case 



we consider, R = y. Under these 
equation (11.3.6) takes the form 



circumstances the differential 



dii 
(11.3.10) 



MATHEMATICAL HYDRAULICS 467 

r*(Uy+D)\Uy+D\ 



dC D 2 

2/ 3 
as can be readily seen. 

It is natural to inquire first of all whether (11.3.10) admits of solu- 
tions which are continuous periodic functions of f since this is the 
general type of motion we seek. There are, however, no such periodic 
and continuous solutions (cf. the previously cited paper of Thomas 
[T.l]) of the equations; in fact, since the right hand side of (11.3.10) 
can be expressed as the quotient of cubic polynomials in y the 
types of functions which arise on integrating it are linear combina- 
tions of the powers, the logarithm, and the arc tangent function and 
one hardly expects to find periodic functions on inverting solutions 
C(y) of this type. This fact, together with observations of roll waves of 
the kind shown in Fig. 11.3.3, leads one to wonder whether there might 
not be discontinuous periodic solutions of (11.3.10) with discontinui- 
ties in the form of bores, which should be fitted in so that the discon- 
tinuity or shock conditions described in sec. 6 of the preceding chap- 
ter * are satisfied. This Dressier shows to be the case; he also gives a 
complete quantitative analysis of the various possibilities. 

The starting point of the investigation is the observation, due to 
Thomas [T.l], that only quite special types of solutions of (11.3.10) 
come in question once the roll wave problem has been formulated in 
terms of a periodic distribution of bores. In fact, we know from Chap- 
ter 10 that the flow relative to a bore must be subcritical behind a 
bore but supercritical in front of it; consequently there must be an in- 
termediate point of depth y , say, (cf. Fig. 11.3.4) where the smooth 
flow has the critical speed, i.e. where 

(11-3.11) („, - Vf = gy v 

since U, the speed of the bore, coincides with the propagation speed 
of the wave. At such a point the denominator on the right hand side 
of (11.3.10) vanishes, since D = (v — U)y, and hence dy\dt, would be 
infinite there — contrary to the observations— unless the numerator 
of the right hand side also vanishes at that point. The right hand side 
can now be written as a quotient of cubic polynomials, and we know 

* The shock conditions were derived in Chapter 10 under the assumption 
that no resistances were present. As one would expect, the resistance terms play 
no role in shock conditions, as Dressier [D.12] verifies in his paper. 



468 WATER WAVES 

that numerator and denominator have y as a common root; it follows 
that a factor y — y can be cancelled and the differential equation 
then can be put in the form 

/ U 2 r 2 \ r 2 yl 

{11.3.12) 



dC y 2 +y<>y+y 2 o 

after a little algebraic manipulation. Since the denominator on the 

right hand side is positive and since we seek solutions for which 

dy/dC is everywhere (cf. Fig. 11.3.4) positive, it follows in particular 

that the quadratic in the numerator must be positive for y = y . This 

leads to the following necessary condition for the formation of roll 

waves 

(11.3.13) 4r 2 <*S r , 

obtained by using (11.3.11 ) and other relations. A practically identical 
inequality was derived by Thomas on the basis of the same type of 
reasoning. The inequality states that the channel roughness, which is 
larger or smaller with r 2 , must not be too great in relation to the steep- 
ness of the channel, and this corroborates observations by Rouse 
[R.10] that roll waves can be prevented by making a channel suffi- 
ciently rough. Dressier also shows in his paper that it is important for 
the formation of roll waves that the friction force for the same rough- 
ness coefficient and velocity should increase when the depth decreases; 
he finds, in fact, that roll waves would not occur if the hydraulic ra- 
dius R in the Chezy formula (11.3.9) were to be assumed independent 
of the depth y. 

Dressier goes on in his paper to show that smooth solutions of 
(11.3.12) can be pieced together through bores in such a way that the 
conditions referring to continuity of mass and momentum across the 
discontinuity are satisfied as well as the inequality requiring a loss 
rather than a gain in energy. For the details of the calculations and a 
quantitative analysis in terms of the parameters, the paper of Dressier 
should be consulted, but a few of the results might be mentioned here. 
Once the values of the slope S and the roughness coefficient r 2 are 
prescribed by the physical situation, and the wave propagation speed 
U is arbitrarily given, there exists a one-parameter family of possible 
roll-waves. As parameter the wave length A, i.e. the distance between 
two successive bores, can be chosen; if this parameter is also fixed, 
the roll wave solution is uniquely determined. A specific solution 



MATHEMATICAL HYDRAULICS 469 

would also be fixed if the time period of the oscillation were to be 
fixed together with one other parameter — the average discharge rate, 
say. Perhaps it is in this fashion that the roll waves are definitely 
fixed in some cases — for example, the roll waves down the spill-way 
of a dam are perhaps fixed by the period of surface waves in the dam 
at the crest of the spill- way. Schonfeld [S.4a] discusses the problem 
from the point of view of stability and arrives at the conclusion that 
only one of the solutions obtained by Dressier would be stable, and 
hence it would be the one likely to be observed. 

11.4. Unsteady flows in open channels. The method of characteristics 

In treating unsteady flows it becomes necessary to integrate the 
nonlinear partial differential equations (11.1.1) and (11.1.6) for pre- 
scribed initial and boundary conditions. It has already been mentioned 
that such problems fall into the same category as the problems treated 
in the preceding chapter, since they are of hyperbolic type in two 
independent variables and thus amenable to solution by the method 
of characteristics. It is true that the equations (11.1.1) and (11.1.6) 
are more complicated than those of Chapter 10 because of the occur- 
rence of the variable coefficient A and of the resistance term, so that 
solutions of the type called simple waves (cf. Ch. 10.3) do not exist for 
these equations. Nevertheless the theory of characteristics is still 
available and leads to a variety of valuable and pertinent observa- 
tions regarding the integration theory of equations (11.1.1) and (11.1. 
6) which are very important. The essential facts have already been 
stated in Chapter 10.2, but we repeat them briefly here for the sake 
of preserving the continuity of the discussion. Our emphasis in this 
chapter is on numerical solutions, which can be obtained by operating 
with the characteristic form of the differential equations, but since we 
shall not actually use the characteristic form for such purposes we 
shall base the discussion immediately following on a special case, al- 
though the results and observations are applicable in the most general 
case. The special case in question is that of a river of constant rectan- 
gular section and uniform slope, with no flow into the river from the 
banks (i.e. q = in (11.1.2) and (11.1.6)). In this case the differential 
equations can be written as follows: 

(11.4.1) v x y +vy x + y t = 0, 

(11.4.2) Vt +vv x + gy x + E = Q. 



470 WATER WAVES 

We have introduced the symbol E for the external forces per unit 

mass: 

(11.4.3) E = — gS + gS f , S = const. 

The term E differs from the others in that it contains no derivatives 

of y or v. 

The theory of characteristics for these equations can be approached 
very directly * in the present special case by introducing a new 
quantity c to replace y, as follows: 

(n.4.4) c = vgy- 

This quantity has great physical significance, since it represents — as 
we have seen in Chapter 10— the propagation speed of small disturb- 
ances in the river. From (11.4.4) we obtain at once the relations 
(H.4.5) 2cc x = gy x , 2cc t = gy t , 

and the differential equations (11.4.1) and (11.4.2) take the form 

(11.4.6) ( ^c x+ v t+ vv x + E = 0, 
\ cv x + 2vc x + 2c t = 0. 

These equations are next added, then subtracted, to obtain the follow- 
ing equivalent pair of equations: 

(11.4.7) 

We observe that the derivatives in these equations now have the form 
of directional derivatives— indeed, to achieve that was the purpose of 
the transformation— so that c and v in the first equation, for example, 
are both subject to the operator (c -f v)djdx -\- d/dt, which means 
that these functions are differentiated along curves in the x, 2-plane 
which satisfy the differential equation dxjdt — c + v. In similar 
fashion, the functions c and v in the second equation are both subject 
to differentiation along curves satisfying the differential equation 
dxjdt = — c -f v. 

It is entirely feasible to develop the integration theory of equations 

(11.4.7) quite generally on the basis of these observations (as is done, 
for example, in Courant-Friedrichs [C.9, Ch. 2] ), but it is simpler, and 
leads to the same general results, to describe it for the special case in 

* For a treatment which shows quite generally how to arrive at the for- 
mulation of the characteristic equations, see Courant-Friedrichs [C.9, Ch. 2]. 



MATHEMATICAL HYDRAULICS 471 

which the resistance force F f is neglected so that the quantity E in 
(11.4.7) is a constant (see (11.4.3)). In this case the equations (11.4.7) 
can be written in the form 

— +(»+c)- \(v+2c+Et)=0, 

dt ox ) 

(11.4.7)! 

j t +(v-c)^\(v-<Zc+Et)=0, 

as one can readily verify. But the interpretation of the operations de- 
fined in (11.4.7)! has just been mentioned: the relations state that the 
functions (v ± 2c + Et) are constant for a point moving through the 
fluid with the velocity (d^ c), or, as we may also put it, for a point 
whose motion in the x, /-plane is characterized by the ordinary dif- 
ferential equations dx/dt = v ± c. That is, we have the following 
situation in the x, /-plane: There are two sets of curves, C x and C 2 , 
called characteristics, which are the solution curves of the ordinary 
differential equations 

C x : — =v-\-c, and 



(11.4.8) 



dt 

dx 
C 2 : — =v— c, 
2 dt 



(11.4.9) 



and we have the relations 

v-\- 2c-\-Et=k 1 = const, along a curve C Y and 
v— 2c -\-Et=k 2 — const, along a curve C 2 . 
Of course the constants k x and k 2 will be different on different curves 
in general. It should also be observed that the two families of charac- 
teristics determined by (11.4.8) are really distinct because of the fact 
that c — \/gy ^ since we suppose that y > 0, i.e. that the water 
surface never touches the bottom. 

By reversing the above procedure it can be seen rather easily that the 
system of relations (11.4.8) and (11.4.9) is completely equivalent to 
the system of equations (11.4.6) for the case of constant bottom slope 
and zero resistance, so that a solution of either system yields a solution 
of the other. In fact, if we set f(x, t) — v + 2c + Et and observe that 
f{x,t) = fcj = const, along any curve x = x(t) for which dx/dt = 
v + c it follows that along such curves 

(11.4.10) /« + /* f£ =f t +(v+c)f x =0. 

at 



472 WATER WAVES 

In the same way the function g(x, t) = v — 2c + Et satisfies relation 
(11.4.11) g t + (v-c)g x = 

along the curves for which dxjdt = v — c. Thus wherever the curve 
families C x and C 2 cover the x, /-plane in such a way as to furnish a 
curvilinear coordinate system the relations (11.4.10) and (11.4.11) 
hold. If now equations (11.4.10) and (11.4.11) are added and the 
definitions of f(x, t) and g(x, t) are recalled it is readily seen that the 
first of equations (11.4.6) results. By subtracting (11.4.11) from 
(11.4.10) the second of equations (11.4.6) is obtained. In other words, 
any functions v and c which satisfy the relations (11.4.8) and (11.4.9) 
will also satisfy (11.4.6) and the two systems of equations are there- 
fore now seen to be completely equivalent. 

As we would expect on physical grounds, a solution of the original 
dynamical equations (11.4.6) could be shown to be uniquely deter- 
mined when appropriate initial conditions (for t = 0, say) and boun- 
dary conditions are prescribed; it follows that any solutions of (11.4. 
8) and (11.4.9) are also uniquely determined when such conditions 
are prescribed since we know that the two systems of equations are 
equivalent. 

At first sight one might be inclined to regard the relations (11.4.8) 
and (11.4.9) as more complicated than the original pair of partial 
differential equations, particularly since the right hand sides of (11.4.8) 
are not known and hence the characteristic curves are also not known. 
Nevertheless, the formulation in terms of the characteristics is quite 
useful in studying properties of the solutions and also in studying 
questions referring to the appropriateness of various boundary and 
initial conditions. In Chapter 10.2 a detailed discussion along these 
lines is given; we shall not repeat it here, but will summarize the con- 
clusions. The description of the properties of the solution is given in 
the x, /-plane, as indicated in Fig. 11.4.1. In the first place, the values 
of v and c at any point P(x, t) within the region of existence of the solu- 
tion are determined solely by the initial values prescribed on the segment 
of the x-axis which is subtended by the two characteristics issuing from P. 
In addition, the two characteristics issuing from P are themselves also 
determined solely by the initial values on the segment subtended by 
them. Such a segment of the #-axis is often called the domain of de- 
pendence of the point P. Correspondingly we may define the range of 
influence of a point Q on the cr-axis as the region of the x, /-plane in 
which the values of v and c are influenced by the initial values assigned 



MATHEMATICAL HYDRAULICS 



473 



to point Q, i.e., it is the region between the two characteristics issuing 
from Q. In Fig. 11.4.1 we indicate these two regions. 




Range of influence of Q Domain of dependence of P 

Fig. 11.4.1. The role of the characteristics 



We are now in a position to understand the role of the charac- 
teristics as curves along which discontinuities in the first and higher 
derivatives of the initial data are propagated, since it is reasonable to 
expect (and could be proved) that those points P whose domains of 
dependence do not contain such discontinuities are points at which 
the solutions v and c also have continuous derivatives. On the other 
hand, it could be shown that a discontinuity in the initial data at a 
certain point does not in general die out along the characteristic 
issuing from that point. Such a discontinuity (or disturbance in the 
water) therefore spreads in both directions over the surface of the 
water with the speed v -f c in one direction and v — c in the other in 
view of the interpretation given to the characteristics through the 
relations (11.4.7)^ Since v is the velocity of the water particles we see 
that c represents quite generally the speed at which a discontinuity 
in a derivative of the initial data propagates relative to the moving 
water. We are therefore justified in referring to the quantity c = \/gy 
as the wave speed or propagation speed. 

We considered above a problem in which only initial conditions, 
and no boundary conditions, were prescribed. In the problems we 
consider later, however, such boundary conditions will occur in the 
form of conditions prescribed at a certain fixed point of the river in 
terms of the time: for example, the height, or stage, of the river might 
be given at a certain station as a function of the time. In other words, 



474 WATER WAVES 

conditions would be prescribed not only along the #-axis of our x, t- 
plane, but also along the /-axis (in general only for t > 0) for a certain 
fixed value of x. The method of finite differences used in Chapter 10.2 
to discuss the initial value problem, with the general result given above, 
can be modified in an obvious way to deal with cases in which bound- 
ary conditions are also imposed. In doing so, it would also become 
clear just what kind of boundary conditions could and should be im- 
posed. For example, in the great majority of rivers— in fact, for all 
in which the flow is subcritical, i.e. such that v is everywhere less than 
the wave speed <\/gy— it is possible to prescribe only one condition 
along the /-axis, which might be either the velocity v or the depth y, 
in contrast with the necessity to impose two conditions along the pr- 
axis. This fact would become obvious on setting up the finite differ- 
ence scheme, and examples of it will be seen later on. 

Finally, it should be stated that the role of the characteristics, and 
also the method of finite differences applied to them could be used 
with reference to the general case of the characteristic equations as 
embodied in the equations (11.4.7) and (11.4.8) in essentially the 
same way as was sketched out above for the system comprised of 
(11.4.8) and (11.4.9) which referred to a special case. In particular, 
the role of the characteristics as curves along which small disturbances 
propagate, and their role in determining the domain of dependence, 
range of influence, etc. remain the same. 

11.5. Numerical methods for calculating solutions of the differential 
equations for flow in open channels 

It has already been stated that while the formulation of our pro- 
blems by the method of characteristics is most valuable for studying 
many questions concerned with general properties of the solutions 
of the differential equations, it is in most cases not the best formula- 
tion to use for the purpose of calculating the solutions numerically. 
That is not to say that the device of replacing derivatives by differ- 
ence quotients should be given up, but rather that this device should 
be used in a different manner. The basic idea is to operate with finite 
differences by using a fixed rectangular net in the x, /-plane, in con- 
trast with the method outlined in Chapter 10.2, in which the net of 
points in the x, /-plane at which the solution is to be approximated is 
determined only gradually in the course of the computation. In the 
latter procedure it is thus necessary to calculate not only the values 



MATHEMATICAL HYDRAULICS 



475 



of the unknown functions v and c, but also the values of the coordinates 
x, t of the net points themselves, whereas a procedure making use of a 
fixed net would require the calculation of v and c only, and it would 
also have the advantage of furnishing these values at a convenient set 
of points. 

However, the question of the convergence of the approximate 
solution to the exact solution when the mesh width of a rectangular 
net is made to approach zero is more delicate than it is when the meth- 
od of characteristics is used. For example, it is not correct, in general, 
to choose a net in which the ratio of the mesh width At along the /-axis 
and the mesh width Ax along the #-axis is kept constant independent 
of the solution: such a procedure would not in general yield approxi- 
mations converging to the solution of the differential equation pro- 
blem. The reason for this can be understood with reference to one of 
the basic facts about the solution of the differential equations which 
was brought out in the discussion of the preceding section. The basic 
fact in question is the existence of what was called there the domain 
of dependence of the solution. For example, suppose the solution were 
to be approximated at the points of the net of Fig. 11.5.1a by advanc- 
ing from one row parallel to the x-slxis to the next row a distance At 
from it. In addition, suppose this were to be done by determining the 
approximate values oft; and c at any point such as P (cf. Fig. 11.5.1b) 



t 




a b 

11.5.1. Approximation by using a rectangular net 



by using the values of these quantities at the nearest three points 
0, 1, 2 in the next line below, replacing derivatives in the two different- 
ial equations by difference quotients, and then solving the resulting 
algebraic equations for the two unknowns v P and c P . It seems reason- 
able to suppose that such a scheme would be appropriate only if P 
were in the triangular region bounded by the characteristics drawn 
from points and 2 to form the region within which the solution is de- 



476 WATER WAVES 

termined solely by the data given on the segment — 2: otherwise it 
seems clear that the initial values at additional points on the #-axis 
ought to be utilized since our basic theory tells us that the initial data 
at some of them would indeed influence the solution at point P. On the 
other hand, the characteristic curves themselves depend upon the 
values of the unknown functions v and c— their slopes, in fact, are 
given (cf. (11.4.8)) by dxjdt = w±c and thus the interval At must be 
chosen small enough in relation to a fixed choice of the interval Ax 
so that the points such as P will fall within the appropriate domains of 
determinacy relative to the points used in calculating the solution at 
P. In other words, the theory of characteristics, even if it is not used 
directly, comes into play in deciding the relative values of At and Ax 
which will insure convergence (for rigorous treatments of these 
questions see the papers by Courant, Isaacson, and Rees [C.ll], and 
by Keller and Lax [K.4]). 

We shall introduce two different schemes employing the method of 
finite differences in a fixed rectangular net of the x, 2-plane. The first 
of these makes use of the differential equations in the form given by 
(11.4.7), and we no longer suppose that the function E is restricted in 
any way. (It might be noted that the slopes of the characteristics as 
given by (11.4.8) are determined by the quantities v _4r c, no matter 
how the function E is defined, and in fact also for the most general 
case of a river having a variable cross section A, etc., and hence we are 
in a position to determine appropriate lengths for the ^-intervals, in 
accord with the above discussion, in the most general case. This is 
also a good reason for working with the quantity c in place of y.) 
At the same time, the calculation is based on assuming that the ap- 



P 

L M R 

I— Ax — H«— Ax — ^ 



r- 



Fig. 11.5.2. A rectangular net 

proximate values of c and v have been calculated at the net points 
L, M, R (cf. Fig. 11.5.2) and that the differential equations are to be 



MATHEMATICAL HYDRAULICS 477 

used to advance the approximate solution to the point P. The differ- 
ential equations to be solved are thus 

(11.5.1) 2{(c + v)c x + c t } + {(c + v)v x + v t } + E = 0, 

(11.5.2) - 2{(- c + z;)c x + c t } + {(- c + ofc + »J + # = 0, 

and the characteristic directions are determined by dxjdt = »ic 
The characteristic with slope v + c we call the forward characteristic, 
and that with slope p — c the backward characteristic. We shall re- 
place the derivatives in the equations by difference quotients which 
approximate the values of the derivatives at the point M. In order to 
advance the values of v and c from the points L, M, R to the point P 
by using (11.5.1) and (11.5.2) it is natural to replace the time deriva- 
tives v t and c t by the following difference quotients 

(1 ,,3) —^. , = ^ 

in both equations. However, in order to insure the convergence of 
the approximations to the exact solution when Ax -> and At -> 
(see Courant, Isaacson, and Rees [C.ll] for a proof of this fact) it is 
necessary to replace the derivatives v x and c x by difference quotients 
which are defined differently for (11.5.1) than for (11.5.2), as follows: 

(11.5.4) v x = ^^- L , c x = ^l^ in (11.5.1), 

Ax Ax 

(11.5.5) v x = ^^, c x = C JL=JH in (11.5.2). 

Ax Ax 

The reason for this procedure is, at bottom, that (11.5.1 ) is an equation 
associated with the forward characteristic, while (11.5.2) is associated 
with the backward characteristic. The coefficients of the derivatives 
and the function E are, of course, to be evaluated at the point M . The 
difference equations replacing (11.5.1) and (11.5.2) are thus given by 

(11.5.6) 2(( Cm+I , m ) C -^-^+ C -^_^ 

( Ax At 

+ {(c M + v M ) ^_ZiL L + ^^H 1 
Ax At 



(11.5.7) -<i\(-c M +v M ) 



C R C M | C P C M 

Ax At 



(- C «+ V M) V± ~^ V -^ fL 



478 WATER WAVES 

We observe that the two unknowns, v P and c P , occur linearly in these 
equations; hence they are easily found by solving the equations. The 
result is 

(11.5.8) Vp = v M + — [(c M +v M )(±v L -iv M +c L -c M ) 

Ax 

— {c M — v M )(iv M — ¥r~ c m+c r )— AxE m ] 9 

(11.5.9) c P = c M +\— [{c M +v M ){\v L -\v M +c L —c M ) 

ZaX 

+( c M—v M )(hv M —iv R —c M -\-c R )]. 

In accordance with the remarks made above, we must also require that 
the ratio of At to Ax be taken small enough so that P lies within the 
triangle formed by drawing lines from L and R in the directions of the 
forward and backward characteristics respectively, i.e. lines with the 
slopes v L + c L at L and v R — c R at R: a condition that is well-de- 
termined since the values of v and c are presumably known at L and R. 

One can now see in general terms how the initial value problem 
starting at t = can be solved approximately: One starts with a net 
along the x-axis with spacing Ax. Since c and v are known at all of 
these points, the values of c and v can be advanced through use of 
(11.5.8) and (11.5.9) to a parallel row of points on a line distant At 
along the 2-axis from the «r-axis. However, the mesh width At must 
be chosen small enough so that the convergence condition is satisfied 
at all net points where new values of v and c are computed. 

We can now see also how to take care of boundary conditions, i.e. 
of conditions imposed at a fixed point (say at the origin, x = 0) as 
given functions of the time. For example, the depth y (corresponding 
to the stage of the river) or the velocity v (which together with the 
cross-section area A fixes the rate of discharge) might be given in 
terms of the time. Initial conditions downstream from this point (i.e. 
for x > 0) might also be prescribed. Suppose, for example, that the 
stage of the river is prescribed at x = 0, i.e. that y(0, t) is known, and 
that the calculation had already progressed so far as to yield values of 
v and c at net points along a certain line parallel to the #-axis and 
containing the points L, M, R, as indicated in Fig. 11.5.3. It is clear 
that the determination of the values of v and c at point P can be ob- 
tained from their values at L, M, R by using (11.5.8) and (11.5.9), 
as in the above discussion of the initial value problem, and similarly 



MATHEMATICAL HYDRAULICS 479 

for points P v P 2 , etc. However, the value of v at Q must be deter- 
mined in a different manner; for this purpose we use the equation 
(11.5.7) with the subscript Q replacing P, L replacing M, and M 
replacing R. Since v M , c M , v R , c R are supposed known, and c Q is also 



Q .p .P, .P 2 .P 3 

L M R 



Fig. 11.5.3. Satisfying boundary conditions 

known since the values of y are prescribed on the /-axis, it follows that 
equation (11.5.7) contains Vq as the only unknown; in fact it is given 
by the equation 

1 



(11.5.10) v Q = v L +At \ — {c L -v L )(2c L -2c M -v L +v M )-E L \ +2(c Q -c L ] 



The reason for using (11.5.7) instead of (11.5.6) is, of course, that the 
points M and Q are associated with the backward characteristic, and 
hence (11.5.2) should be used to approximate the ^-derivatives at 
point L (where the differential equations are replaced by difference 
equations). It is quite clear that the same general procedure could be 
used to calculate Cq if the values of v had been assumed given along 
the /-axis. If, on the other hand, we had a boundary condition on the 
right of our domain instead of on the left, as above, we could make use 
of (11.5.6) for the forward characteristic as a basis for obtaining the 
formula for advancing the solution along the /-axis. 

The above discussion would seem to imply that under all circum- 
stances only one boundary condition could be imposed — that is, that 
either v or c could be prescribed at a fixed point on the river, but not 
both — since prescribing one of these quantities leads to a unique de- 
termination of the other. This is, indeed, true in any ordinary river, 
but not necessarily in all cases. In fact, we made a tacit assumption 
in the above discussion, i.e. that of the two characteristics issuing 
from any point of the /-axis only the forward characteristic goes into 



480 



WATER WAVES 



the region x > to the right of the £-axis, and this in turn implies that 
v -\- c and v — c, which fix the slopes of the characteristics, are op- 
posite in sign. The physical interpretation of this is that the value of 
v (which is positive here) must be less than c = \/gy, i.e. that the 
flow must be what is called tranquil, or subcritical.* Otherwise, as 
we see from Fig. 11.5.4, we should expect to determine the values of 
v and c at points close to and to the right of the £-axis, say at K, by 




Fig. 11.5.4. A case of super-critical flow 

utilizing values for both v and c along the segment LQ, its domain of 
dependence. The scheme outlined above would therefore have to be 
modified in a proper way under such circumstances. One sees, how- 
ever, how useful the theory based on the characteristics can be even 
though no direct use of it is made in the numerical calculations (aside 
from decisions regarding the maximum permissible size of the ^-inter- 
val). 

The procedure sketched out above, while it is recommended for use 



L M R 



At 



I*— Ax 



Fig. 11.5.5. A staggered net 
* In gas dynamics the flow in an analogous case would be called subsonic. 



MATHEMATICAL HYDRAULICS 



481 



when a boundary condition is to be satisfied, is not always the best 
one to use for advancing the solution to such points as P, P l3 P 2 , . . . 
in Fig. 11.5.3. For such "interior points" a staggered rectangular net, 
as indicated in Fig. 11.5.5, and a difference equation scheme based on 
the original differential equations (11.4.6) may be preferable (cf. 
Keller and Lax [K.4] for a discussion of this scheme). The equations 
(11.4.6) were 

( 2cc x + v t + vv x + E = 0, 
(11.5.11) 



cv x + 2vc x + 2c t 



0. 



The values v M and c M at the mid-point M (which is, however, not a 
net point) of the segment LR are defined by the averages: 



(11.5.12) 



v L +v R 



c L +c R 



M 



after which the derivatives at M are approximated in a quite natural 
way by the difference quotients 



(11.5.13; 



Vr~ 


-v L 


A 


X 


v P - 


-v M 



At 



C R~ C L 

Ax 

C P — C M 

~~aT 



Upon substitution of these quantities into (11.5.11), evaluation of the 
coefficients c, v, and E at point M, and subsequent solution of the 
two equations for v P and c P , the result is 



(11.5.14; 



v P = v M + — [2(c L -c R )c M +(v L -v R )v M -AxE M l 

Ax 

c P = c M -\r\— [%(c L -c R )v M + (v L -v R )c M ]. 

Ax 



As we see on comparison with (11.5.8) and (11.5.9 \ these equations 
are simpler than the earlier ones. The criterion for convergence re- 
mains the same as before, i.e. that P should lie within a triangle formed 
by the segment LR and the two characteristics issuing from its 
ends. 



482 WATER WAVES 

11.6. Flood prediction in rivers. Floods in models of the Ohio River 
and its junction with the Mississippi River 

The theory developed in the preceding sections can be used to make 
predictions of floods in rivers on the basis of the observed, or estimat- 
ed, flow into the river from its tributaries and from the local run-off, 
together with the state of the river at some initial instant. Hydraulics 
engineers have developed a procedure, called flood-routing, to accom- 
plish the same purpose. The flood-routing procedure can be deduced 
as an approximation in some sense to the solution of the basic differ- 
ential equations for flow in open channels (cf. the article by B. R. 
Gilcrest in the book by Rouse [R.ll] ), but it makes no direct use of the 
differential equations. However, the flood-routing procedure in ques- 
tion seems not to give entirely satisfactory results in cases other than 
that of determining the progress of a flood down a long river — for 
example, the problem of what happens at a junction, such as that of 
the Ohio and Mississippi Rivers, or the problem of calculating the 
transient effects resulting from regulation at a dam, such as the 
Kentucky dam at the mouth of the Tennessee River, seem to be diffi- 
cult to treat by methods that are modifications of the more or less 
standard flood-routing procedures. Even for a long river like the 
Ohio, the usual procedure fails occasionally to yield the observed river 
stages at some places. On the other hand, the basic differential equa- 
tions for flow in open channels are in principle applicable in all cases 
and can be used to solve the problems once the appropriate data de- 
scribing the physical characteristics of the river and the appropriate 
initial and boundary conditions are known. 

The idea of using the differential equations directly as a means of 
treating problems of flow in open channels is not at all new. In fact, 
it goes back to Massau [M.5] as long ago as 1889. Since then the idea 
has been taken up by many others (mostly in ignorance of the work 
of Massau)— for example, by Preiswerk [P. 16], von Karman [K.2], 
Thomas [T.2], and Stoker [S.19]. Thomas, in particular, attacked the 
flood-routing problem in his noteworthy and pioneering paper and 
outlined a numerical procedure for its solution based on the idea of us- 
ing the method of finite differences. However, his method is very la- 
borious to apply and would also not necessarily furnish a good 
approximation to the desired solution even if a large number of 
divisions of the river into sections were to be taken. In general, the 
amount of numerical work to be done in a direct integration of the 



MATHEMATICAL HYDRAULICS 483 

differential equations looked too formidable for practical purposes 
until rather recently. 

During and since the late war new developments have taken place 
which make the idea of tackling flood prediction and other similar 
problems by numerical solution of the relevant differential equations 
quite tempting. There have been, in fact, developments in two differ- 
ent directions, both motivated by the desire to solve difficult problems 
in compressible gas dynamics: 1) development of appropriate nu- 
merical procedures— for the most part methods using finite differences 
—for solving the differential equations, and 2) development of com- 
puting machines of widely varying characteristics suitable for carry- 
ing out the numerical calculations. As we have seen, the differential 
equations for flood control problems are of the same type as those for 
compressible gas dynamics, and consequently the experience and cal- 
culating equipment developed for solving problems in gas dynamics 
can be used, or suitably modified, for solving flood control problems. 

In carrying out such a study of an actual river it is necessary to 
make use of a considerable bulk of observational data— cross-sections 
and slopes of the channels, measurements of river depths and dis- 
charges as functions of time and distance down the river, drainage 
areas, observed flows from tributaries, etc. — in order to obtain the 
information necessary to fix the coefficients of the differential equa- 
tions and to fix the initial and boundary conditions. This is a task 
with many complexities. For the purposes of this book it is more 
reasonable to carry out numerical solutions for problems which are 
simplified versions of actual problems. The present section has as its 
purpose the presentation of the solutions in a few such special cases, 
together with an analysis of their bearing on the concrete problems 
for actual rivers. In any case, the general methods for an actual river 
would be the same — there would simply be greater numerical compli- 
cations. 

The simplified models chosen correspond in a rough general way 
(a) to two types of flow for the Ohio River and (b) to the Ohio and 
Mississippi Rivers at their junction. Rivers of constant slope, with 
rectangular cross-sections having a uniform breadth, and with con- 
stant roughness coefficients are assumed. In this way differential 
equations with constant coefficients result. The values of these quan- 
tities are, however, taken to correspond in order of magnitude with 
those for the actual rivers. In the model of the Ohio, for example, the 
slope of the channel was assumed to be 0.5 ft /mile, the quantity n 



484 WATER WAVES 

(the roughness coefficient in Manning's formula) was given the value 
0.03, and the breadth of the river was taken as 1000 feet. It is assumed 
that a steady uniform flow with a depth of 20 ft existed at the initial 
instant t = 0, and that for t > the depth of the water was increased 
at a uniform rate at the point x = from 20 ft to 40 ft within 4 
hours and was then held fixed at the latter value. (These depths are 
the same as for the problem of a steady progressing wave treated in 
sec. 11.2 above.) The problem is to determine the flow downstream, 
i.e. the depth y and the flow velocity v as functions of x (for x > 0) 
and t. 

The methods used to obtain the solution of this problem of a flood in 
a model of the Ohio River, together with a discussion of the results, 
will be given in detail later on in this section. Before doing so, a few 
general remarks and observations about them should be made at this 
point. In the first place, it was found possible to carry out the solution 
numerically by hand computation over a considerable range of dis- 
tances and times (values at 900 net points in the x, 2-plane were de- 
termined by finite differences), and this in itself shows that the 
problems are well within the capacity of modern calculating equip- 
ment. It might be added that the special case chosen for a flood in the 
Ohio was one in which the rate of rise at the starting point upstream 
was extremely high (5 feet per hour, in comparison with the rate of 
rise during the flood of 1945— one of the biggest ever recorded in the 
Ohio— which was never larger than 0.7 feet per hour at Wheeling, 
West Virginia), so that a rather severe test of the finite difference 
method was made in view of the rapid changes of the basic quantities 
in space and time. The decisive point in estimating the magnitude of 
the computational work in using finite differences is the number of 
net points needed; for a river such as the Ohio it is indicated that an 
interval Ax of the order of 10 miles along the river and an interval At 
of the order of 0.3 hours in time in a rectangular net in the x, 2-plane 
will yield results that are sufficiently accurate. (Of course, a problem 
for the Ohio in its actual state involves empirical coefficients in the 
differential equations and other empirical data, which must be coded 
for calculating machines, but this would have no great effect on these 
estimates for Ax and might under extreme flood conditions reduce At 
by a factor of 1/2.) 

As we know from sec. 11.3 above, there is a case in which an exact 
solution of the differential equations is known, i.e. the case of a 
steady progressing wave with two different depths at great distances 



MATHEMATICAL HYDRAULICS 485 

upstream and downstream. The exact solution obtained in sec. 11.3 
for the case of a wave of depth 20 ft far downstream and 40 ft far 
upstream was taken as furnishing the initial conditions at / — for 
a wave motion in the river. With the initial conditions prescribed in 
this way the finite difference method was used to determine the mo- 
tion at later times; of course the calculation, if accurate, should fur- 
nish a wave profile and velocity distribution which is the same at 
time t as at the initial instant t = except that all quantities are dis- 
placed downstream a distance Ut, with U the speed of the steady 
progressing wave. In this way an opportunity arises to compare the 
approximate solution with an exact solution. In the present case the 
phase velocity U is approximately 5 mph. Interval sizes of Ax = 5 
miles in a "staggered" finite difference scheme (cf. equations (11.5.14)) 
with At = .08 hr were taken and a numerical solution was worked 
out. We report the results here. After 12 hours, the calculated values 
for the stage y agreed to within .5 per cent with the exact values. 
The discharge and the velocity deviated by less than .8 per cent 
from the exact values. 

One of the valuable insights gained from working out the solution 
of the flood problem in a model of the Ohio was an insight into the 
relation between the methods used by engineers— for example, by 
the engineers of the Ohio River Division of the Corps of Engineers in 
Cincinnati — for predicting flood stages, and the methods explained 
here, which make use of the basic differential equations. At first sight 
the two methods seem to have very little in common, though both, in 
the last analysis, must be based on the laws of conservation of mass 
and momentum; indeed, in one important respect they even seem to 
be somewhat contradictory. The methods used in engineering prac- 
tice (which make no direct use of our differential equations) tacitly 
assume that a flood wave in a long river such as the Ohio propagates 
only in the downstream direction, while the basic theory of the dif- 
ferential equations we use tells us that a disturbance at any point in 
a river flowing at subcritical speed (the normal case in general and 
always the case for such a river as the Ohio) will propagate as a wave 
traveling upstream as well as downstream. Not only that, the speed 
of propagation of small disturbances relative to the flowing stream, as 
defined by the differential equations, is \/gy for small disturbances 
and this is a good deal larger (by a factor of about 4 in our model of 
the Ohio) than the propagation speed used by the engineers for their 
flood wave traveling downstream. There is, however, no real dis- 



486 WATER WAVES 

crepancy. The method used by the engineers can be interpreted as a 
method which yields solutions of the differential equations, with cer- 
tain terms neglected, that are good approximations (though not under 
all circumstances, it seems) to the actual solutions in some cases, 
among them that of flood waves in a river such as the Ohio. The 
neglect of terms in the differential equations in this approximate 
theory is so drastic as to make the theory of characteristics, from 
which the properties of the solutions of the differential equations were 
derived here, no longer available. The numerical solution presented 
here of the differential equations for a flood wave in a model of the 
Ohio yields, as we have said, a wave the front of which travels down- 
stream at the speed \/gy; but the amplitude of this forerunner is 
quite small,* while the portion of the wave with an amplitude in the 
range of practical interest is found by this method to travel with 
essentially the same speed as would be determined by the engineers' 
approximate method. What seems to happen is the following: small 
forerunners of a disturbance travel with the speed \/gy relative to the 
flowing stream, but the resistance forces act in such a way as to de- 
crease the speed of the main portion of the disturbance far below the 
values given by \/gy, i.e. to a value corresponding closely to the speed 
of a steady progressing wave that travels unchanged in form. (One 
could also interpret the engineering method as one based on the as- 
sumption that the waves encountered in practice differ but little from 
steady progressing waves). As we shall see a little later, our unsteady 
flow tends to the configuration of a steady progressing wave of depth 
40 ft upstream and 20 ft downstream. 

This analysis of the relation between the methods discussed here 
and those commonly used in engineering practice indicated why it 
may be that the latter methods, while they furnish good results in 
many important cases, fail to mirror the observed occurrences in other 
cases. For example, the problem of what happens at a junction of two 
major streams, and various problems arising in connection with the 
operation of such a dam as the Kentucky Dam in the Tennessee River 
seem to be cases in which the engineering methods do not furnish 
accurate results. These would seem to be cases in which the motions 
of interest depart too much from those of steady progressing waves, 
and cases in which the propagation of waves upstream, is as vital as the 
propagation downstream. Thus at a major junction it is clear that 

* In an appendix to this chapter an exact statement on this point is made. 



MATHEMATICAL HYDRAULICS 487 

considerable effects on the upstream side of a main stream are to be 
expected when a large flow from a tributary occurs. In the same way, 
a dam in a stream (or any obstruction, or change in cross-section, etc.) 
causes reflection of waves upstream, and neglect of such reflections 
might well cause serious errors on some occasions. 

The above general description of what happens when a flood wave 
starts down a long stream — in particular, that it has a lengthy front 
portion which travels fast, but has a small amplitude, while the main 
part of the disturbance moves much more slowly — has an important 
bearing on the question of the proper approach to the numerical solu- 
tion by the method of finite differences. It is, as we shall see shortly, 
necessary to calculate — or else estimate in some way — the motion up 
to the front of the disturbance in order to be in a position to calculate 
it at the places and times where the disturbances are large enough to 
be of practical interest. This means that a large number of net points 
in the finite difference mesh in the x, ^-plane lie in regions where the 
solution is not of much practical interest. Since the fixing of the solu- 
tion in these regions costs as much effort as for the regions of greater 
interest, the differential equation method is at a certain disadvantage 
by comparison with the conventional method in such a case. However, 
it is possible in simple cases to determine analytically the character 
of the front of the wave and thus estimate accurately the places and 
times at which the wave amplitude is so small as to be negligible; 
these regions can then be regarded as belonging to the regions of the 
x, 2-plane where the flow is undisturbed, with a corresponding re- 
duction in the number of net points at which the solutions must be 
calculated. A method which can be used for this purpose has been 
derived by G. Whitham and A. Troesch, and a description of it is 
given in an appendix to this chapter. If a modern high speed digital 
computer were to be used to carry out the numerical work, however, 
it would not matter very much whether the extra net points in the 
front portion of the wave were to be included or not: many such 
machines have ample capacity to carry out the necessary calculations. 

We proceed to give a description of the calculations made for our 
model of the Ohio, including a discussion of various difficulties which 
occurred for the flood wave problem near the front of the disturbance, 
and particularly at the beginning of the wave motion (i.e. near x = 0, 
t = 0), and an enumeration of the features of the calculation which 
must play a similar role in the more complicated cases presented by 
rivers in their actual state. This will be followed by a description of 



488 WATER WAVES 

the method used and the calculations made for a problem simulating 
a flood coming down the Ohio and its effect on passing into the 
Mississippi. This problem and its solution give rise to further general 
observations which will be made later on. 
The differential equations to be solved are 

( 2cc x + v. + vv x + E = 0, 
11.6.1 x l x 

\ 2c t + 2vc x + cv x = 0, 

with v(x, t) the velocity, and c = Vgy the propagation speed of 
small disturbances. The assumption of a uniform cross-section and 
the assumption that no flow over the banks occurs (i.e. q = in the 
basic differential equations (11.1.1) and (11.1.6)) have already been 
used. The quantity E is given by 

E = - gS + gS„ 

with S the slope of the river bed and S f , the friction slope, given by 
Manning's formula 

a, ""*■" 



M 



2y 

1 + i 



Here we assume the channel to be rectangular with breadth B. 

The numerical data for the problem of a flood in a model of the 
Ohio River are as follows. For the slope S a value of 0.5 ft/mi was 
chosen, and B is given the value 1000 ft. For y a value of 2500 was 
taken (in foot-sec units), corresponding to a value of Manning's 
constant n (in the formula y = (1.49/w) 2 ) of 0.03. The special pro- 
blem considered was then the following: At time t = 0, a steady flow 
of depth 20 ft is assumed. At the "headwaters" of the river, corres- 
ponding to x = 0, we impose a linear increase of depth with time which 
brings the level to 40 ft in 4 hours. For subsequent times the level of 
40 ft at x = is maintained. The initial velocity of the water cor- 
responding to a uniform flow of depth y = 20 ft is calculated from 
S f = S to be 

v = 2.38 mph; 

the propagation speed of small disturbances corresponding to the 
depth of 20 ft is 



MATHEMATICAL HYDRAULICS 



489 



c o = Vgy = 17 - 3 m P h - 

The problem then is to determine the solution of (11.6.1) for v(x, t), 
c(x, t) for all later times t ^ along the river x ^ 0. Figures 11.6.1 
and 11.6.2 present the result of the computation in the form of stage 
and discharge curves plotted as functions of distance along the river 
at various times. 

In order to indicate how the solution was calculated it is conven- 
ient to refer to diagrams in the (x, t) plane given by Figs. 11.6.3 
and 11.6.4. According to the basic theory, we know that for x ^ 
(flo + c o) t = 19-7*, called region O in Fig. 11.6.3, the solution is given 



> y 



10 




535^553553^ 



Legend 
t - time in hours after start of flood 
y - stage in feet 
x - distance along Ohio in miles 



^^m^m^^j^^^mm^^m^^^^W: 



20 



40 



60 



80 



100 120 



Fig. 11.6.1. Stage profiles for a flood in the Ohio River 



by the unchanged initial data, v(x, t) 



c(x, t) = c (since the 



forerunner of the disturbance travels at the speed v + c = 19.7 mph). 

Experiments were made with various interval sizes and finite 
difference schemes in order to try to determine the most efficient way 
to calculate the progress of the flood. We proceed to describe the 
various schemes tried and the regions in which they were used on the 
basis of Figs. 11.6.3 and 11.6.4. 

Region I, ^ x ^ 19.7*, ^ t ^ .4. Quite small intervals of 



490 



WATER WAVES 



300. 



260 



220 



Legend 
t= time in hours after start of flood 
Q= discharge in 1000 c.f.s. 
x - distance along Ohio in miles 




20 40 60 80 100 120 140 

Fig. 11.6.2. Discharge records for a flood in the Ohio River 



160 



1.25 




Fig. 11.6.3. Regions in which various computational methods were tried 

Ax = 1 mile and At = .048 hours were required owing to the sudden 
increase of depth at x = 0, t = 0. The finite difference formulas given 
above in equations (11.5.8), (11.5.9) were used. 

In Region II, ^ x ^ 19.7*, .4 ^ t ^ .7, with Ax = 1 mile, 



MATHEMATICAL HYDRAULICS 



491 




Legend 

t s time in hours 
x = distance in miles 



i 
20 



Fig. 11.6.4. Net points used in the finite difference schemes 



At = .024 hr, the "staggered" scheme was used. The formulas for this 
scheme have been given above in equations (11.5.14). In order to 
calculate v(0, t), the velocity at the upstream boundary of the river, 
the formula associated with the backward characteristic, namely 
equation (11.5.10), has to be used twice in succession: for the triangles 
FBM and MRP (cf. Figs. 11.5.3 and 11.6.5). The values c B and v B 
are simply determined by linear interpolation from the values at the 
points F and G. 



492 WATER WAVES 

Region III, ^^5, .7 ^t ^ 1.25, with Ax = 1 mile, At = .024 hr. 
The same procedure was used as in Region II. 

Region IV, 5 ^ x ^ 19.7*, .7 ^ t < 1.25, with Ax = 2 miles, 
Zl< = .048 hr. The values at the boundary between Regions III and 



ft 
p 

M .R 

IF X B 



x 
Fig. 11.6.5. Net point arrangement used at boundary in "staggered" scheme 

IV were obtained by linear interpolation from the neighboring values. 

Other quantities were computed by the "staggered" scheme as in 

Regions II and III. 

Region V, ^ x ^ Ut, 1.25^*^10, Ax = 5 miles, At = .11 hr. 

U represents a variable speed which marks the downstream end of 

what might be called the observable disturbance (U & 10 mph). 

That is, by using an expansion scheme (see the appendix to this 

chapter) we obtain the solution in 

Region VI, defined by Ut 5^ x ^ 19.7/, back of the forerunner 

of the disturbance, in which the flow is essentially undisturbed 

for all practical purposes. The expansion valid near the front of the 

wave and referred to above was used to calculate the various quantities 

in Region VI, and a staggered scheme was used to compute the values 

in Region V. 

A number of conclusions reached on the basis of the experience 

gained from these calculations of a flood in a model of the Ohio River 

can be summarized as follows: 

(a) The rate of rise of the flood — 5 feet per hour — is extreme, and 
such a case exaggerates the way in which errors in the finite 
difference methods are propagated. For example, slight inaccu- 
racies at the head, x = 0, were found to develop upon increasing 
the size of the Ax interval. In spite of the exceptionally high rate 
of rise of the flood, the fluctuations created by using finite dif- 
ference methods were damped out rather strongly (in about 
8 — 10 time steps). It is possible to control these inaccuracies 



MATHEMATICAL HYDRAULICS 



493 



simply by using small interval sizes. The process by which the 
small errors of the finite difference scheme are caused to die out 
may be described as follows: A value of v which is too large 
produces a correspondingly larger friction force which slows down 
the motion and produces at a later time a smaller velocity. The 
lower velocity in a similar way then operates through the resistance 
to create a larger velocity and the process repeats in an oscillatory 
fashion with a steady decrease in the amplitude of variation. 

(b) The accuracy of our computation (as a function of the interval 
size) was checked by repeating the calculation for two different 
interval sizes over the same region in space and time. 

(c) A linearized theory of wave propagation, obtained by assuming 
a small perturbation about the uniform flow with 20 ft depth, is 
easily obtained, and the problem was solved using such a theory. 
However, it does not give an accurate description of the solution 
of our problem. It was found that the stage was predicted too low 
by the linear theory by as much as 2 feet after only 2 hours — a 
very large error. 

(d) It would be convenient to be in possession of a safe estimate for 
the maximum value of the particle velocity, in order to select an 
appropriate safe value for the time interval At, since we must 
have At ^ Axj(v -f c) in order to make sure that the finite dif- 
ference scheme converges. The calculations in our special case 
indicate that this may not be easy to obtain in a theoretical way, 
since the maximum velocity at x = 0, for example, greatly ex- 
ceeds its asymptotic value, as indicated in Fig. 11.6.6. In a 



*v(0,t) 

mph 



5.4- 




2.4. 



velocity for 
40 ft. steady flow 



hours 



Fig. 11.6.6. Water velocity obtained at "head" of river 



computation for an actual river, however, no real difficulty is 
likely to result, since c is in general much larger than v and is 
determined by the depth alone. 



494 



WATER WAVES 



(e) 



As was already indicated above, the curves of constant stage 
turn out to have slopes which are closer to 5 mph (the speed with 
which a steady progressing flow, 40 ft upstream and 20 ft down- 
stream, moves) than they are to the 19.7 mph speed of pro- 
pagation of small disturbances. This is shown by Fig. 11.6.7. 



, 


I t hours 






region of 

practically 

undisturbed 


10. 


/ / 






flow 




ze'A2y28'y 


'24/ 


yio' 


-<H9.7t 


5, 


0^ 




^v 


^ 




5mph-slope 


19.7 mph-slope 
19-7= v f^ 













50 



100 



x miles 



Fig. 11.6.7. Curves of constant stage — comparison with first characteristic and 
steady progressing flow velocity 

The region of practically undisturbed flow (determined by an 
expansion about the "first" characteristic x =- 19.7/, for which 
see the appendix to this chapter) is shown above. In an actual 
river, we would of course expect the local runoff discharges and the 
non-uniform flow conditions to eliminate largely the region of 
practically undisturbed flow. For this reason it is not feasible 
to use analytic expansion schemes as a means of avoiding 
computational labor. 
We turn next to our model of the junction of the Ohio and Missis- 
sippi Rivers and the problem of what happens when a flood wave 
comes down the Ohio and passes through the junction.* The physical 
data chosen are the same as were used above in sec. 11.2 in discussing 
the problem of a steady flow at a junction. 

We suppose the upstream side of the Mississippi to be identical 
with the Ohio River — i.e. that it has a rectangular cross-section 
1000 ft wide, a slope of .5 ft/mile, and that Manning's constant n has 
the value .03. The downstream Mississippi is also taken to be rectan- 
gular, but twice as wide, i.e. 2000 ft in width, Manning's constant is 
again assumed to have the value .03, but the slope of this branch is 
given the value .49 ft/mile. This modification of the slope was made 

* The analogous problem in gas dynamics would be concerned with the pro- 
pagation of a wave at the junction of two pipes containing a compressible gas. 



MATHEMATICAL HYDRAULICS 



495 



in order to make possible an initial solution corresponding to a uni- 
form flow of 20 ft depth in all three branches. (Such a change is 
necessary in order to overcome the decrease in wetted perimeter 
which occurs on going downstream through the junction.) Figure 
11.6.8 shows a schematic plan of the junction. The concrete problem 
to be solved is formulated as follows. A flood is initiated in the Ohio 




[23 

\Upstream\ / 

ississippi 



1000 V 1000 



L3] 

Downstream 
Mississippi 



Fig. 11.6.8. Schematic plan of junction 

at a point 50 miles above the junction by prescribing a rise in depth of 
the stream at that point from 20 ft to 40 ft in 4 hours — in other 
words, the same initial and boundary conditions were assumed as for 
the case of the flood in the Ohio treated in detail above. After about 
2.5 hours the forerunner, or front, of the wave in the Ohio caused by 
the disturbance 50 miles upstream reaches the junction; up to this 
instant nothing will have happened to disturb the Mississippi, and 
the numerical calculations made above for the Ohio remain valid 
during the first 2.5 hours. Once the disturbance created in the Ohio 
reaches the junction, it will cause disturbances which travel both 
upstream and downstream in the Mississippi, and of course also a 
reflected wave will start backward up the Ohio. The finite difference 
calculations therefore were begun in all three branches from the 
moment that the junction was reached by the forerunner of the Ohio 
flood, and the solution was calculated for a period of 10 hours. 

We proceed to describe the method of determining the numerical 
solution. Let v {1)> c (1) , v {2)> c (2) , u (3) , c (3) represent the velocity v and the 
propagation speed c for the Ohio, upstream Mississippi, and down- 



496 



WATER WAVES 



stream Mississippi, respectively. A "staggered" scheme was used with 
intervals Ax = 5 miles and At = .17 hr as indicated in Fig. 11.6.9. 
The junction point is denoted by x = 0, the region of the Ohio and 



h t 



X • X 

y 

X -P x 

.L xM .R 

.K X A .F X B .G 



Ohio [|] 



Downstream Mississippi C31 



Upstream Mississippi E2J Junction 



Fig. 11.6.9. Junction net point scheme 

the upstream Mississippi are represented by x ^ 0, while the down- 
stream Mississippi is described for x ^ 0. The time t = 2.5 hrs, as 
explained above, corresponds to the instant that the forerunner of 
the flood reaches the junction. 

The values of the quantities v and c at the junction were determined 
as follows: Assume that the values of v and c have been obtained at all 
net points for times preceding that of the boundary net point P, which 
represents a point at the junction. We use at this point the relations 



(i) 



(2) 



(3)' 



since c = ^/gy and the water level is the same in the three branches at 
the junction. In addition, we have 



2/(D^(i) + V( 



2)^(2) 



2 2/(3)^ 



3)^(3)' 



since what flows into the junction from the upstream side of the Mis- 
sissippi and from the Ohio must flow out of the junction into the down- 
stream branch of the Mississippi. If the values of v and c were known 
at the point M in Fig. 11.6.9 in the respective branches of the rivers, 
we could find the values at P from equation (11.5.6) for the Ohio and 
the upstream side of the Mississippi, and equation (11.5.7) for the 



MATHEMATICAL HYDRAULICS 497 

downstream side of the Mississippi. We rewrite the equations for 
convenience, as follows: 

c p(d = c p(2) = c p(3)> ( with c = Vgy)> 

Vp { i) + V P(2) = 2 v P(3) , (since y {1) = y {2) = y (3) ), 

(ii.6. 2)j *[**yr£m +(c MW +v MW ) < c w>- c *«> 



At " myj> " ±KJ " Ax 



I V P(j)— V M( 

At 



^+(%,,+^))("^^)] +E M{j) =0, j=l, 2, 



and 



( „.6.2) 3 - a l e jnzpm +(^ ( 3 ) - C M ( 3)) (Cg(8) 7 Cjf(8)) 

{ At Ax 

'P(3) — V M(S) | / , n v ( V R(3)~ V M(3) 



At 



+ (^(3)-^ ( 3)) ( ^ (3) ?' M(3)) +^M ( 3)=0, 

Ax J 



The above system of six linear equations determines uniquely the 
values z; (1) , c (1) , v {2) , c {2) , v {3) , c (3) at P in terms of their values at the 
preceding points L, M and R. The equations can be solved explicitly. 
The values of the relevant quantities at M are determined in the same 
way from the preceding values at A, F and B. The values at A and B 
are determined by interpolation between the neighboring points 
(K, F) and (F, G) respectively (see Fig. 11.6.9). Of course, it is ne- 
cessary to treat the motions in each of the branches away from the 
junction by the same methods as were described for the problem of 
the Ohio treated above, and this is feasible once the values of v and c 
have been obtained at the junction. 

The results of the calculations are shown in Fig. 11.6.10, which 
furnishes the river profiles, i.e. the depths as functions of the location 
in each of the three branches, for times t = 0, 2.5, 4, and 10 hours 
after the beginning of the flood 50 miles up the Ohio. The curves for 
t = oo are those for the steady flow which was calculated above in 
sec. 11.2 (cf. Fig. 11.2.3). The calculations indicate that the unsteady 
flow does tend to the steady flow as the time increases. Another no- 
ticeable effect is the backwater effect in the upper branch of the 
Mississippi. For example, the stage is increased by about 2 feet at a 
point in the Mississippi 20 miles above the junction and 7.5 hours after 
the flood wave from the Ohio first reaches the junction. 



498 



WATER WAVES 



It might be mentioned that the forerunners of the flood in all three 
branches were computed by using the expansion scheme which is 
explained in the appendix to this chapter. 



x = distance in miles 

measured from junction 

y = stage measured in feet 

t= time in hours offer start 
of flood 



40'- r 




1 1 i 1 1 r 1 1 i 1 1 > 1 1 1 1 1 1 1 1 1 \ 1 1 n i r } 1 1 1 1 n i i n 1 1 1 i i 

Fig. 11.6.10 River profiles for the junction 



11.7. Numerical prediction of an actual flood in the Ohio, and at its 
junction with the Mississippi. Comparison of the predicted with 
the observed floods 

The methods for numerical analysis of flood wave problems in 
rivers developed above and applied to simplified models of the Ohio 
and its junction with the Mississippi have been used to predict the 
progress of a flood in the Ohio as it actually is, and likewise to predict 
the progress of a flood coming from the Ohio and passing through the 
junction with the Mississippi. The data for the flood in the Ohio were 
taken for the case of the big flood of 1945, and predictions were made 
numerically for periods up to sixteen days for the 400-mile long 
stretch of the Ohio extending from Wheeling. West Virginia, to 
Cincinnati, Ohio. For the flood through the junction, the data for 
the 1947 flood were used, and predictions were made in all three 
branches for distances of roughly 40 miles from the junction along 



MATHEMATICAL HYDRAULICS 499 

each branch. In each case the state of the river, or river system, was 
taken from the observed flood at a certain time t = 0; for subsequent 
times the inflows from tributaries and the local run-off in the main 
river valley were taken from the actual records, and then the differ- 
ential equations were integrated numerically with the use of the 
UNIVAC digital computer in order to obtain the river stages and dis- 
charges at future times. The flood predictions made in this way were 
then compared with the actual records of the flood. 

A comparison of observed with calculated flood stages will be given 
later on; however, it can be said in general that there is no doubt that 
this method of dealing with flood waves in rivers is entirely feasible 
since it gives accurate results without the necessity for unduly large 
amounts of expensive computing time on a machine such as the 
UNIVAC. For example, a prediction for six days in the 400-mile 
stretch of the Ohio requires less than three hours of machine time. 
This amount of calculating time— which is anyway not unreasonably 
large — could almost certainly be materially reduced by modifying 
appropriately the basic methods; so far, no attention has been given 
to this aspect of the problem, since it was thought most important 
first of all to find out whether the basic idea of predicting floods by 
integrating the complete differential equations is sound. The fact that 
such problems can be solved successfully in this way is, of course, a 
matter of considerable practical importance from various points of 
view. For example, this method of dealing with flood problems in 
rivers is far less expensive than it is to build models of a long river or 
a river system, and it appears to be accurate. Actually, the two 
methods — empirically by a model, or by calculation from the theory 
— are in the present case basically similar, since the models are really 
huge and expensive calculating machines of the type called analogue 
computers, and the processes used in both methods are at bottom the 
same, even in details. An amplification of these remarks will be made 
later on. 

It would require an inordinate amount of space in this book to deal 
in detail with the methods used to convert the empirical data for a 
river into a form suitable for computations of the type under discussion 
here, and with the details of coding for the calculating machine; for 
this, reference is made to a report [1.4]. Instead, only a brief outline 
of the procedures used will be given here. 

In the first place, it is necessary to have records of past floods with 
stages up to the maximum of any to be predicted. It would be ideal 



500 WATER WAVES 

to have records of flood stages and discharges (or, what comes to the 
same thing, of average velocities over a cross-section) at points closely 
spaced along the river— at ten mile intervals, say. Unfortunately, 
measurements of this kind are available only at much wider inter- 
vals * — of the order of 50 to 80 miles or more — even in the Ohio 
River, for which the data are more extensive than for most rivers in 
the United States. From such records, it is possible to obtain the co- 
efficient of the all-important resistance term in the differential equa- 
tion expressing the law of conservation of momentum. This coefficient 
depends on both the location of the point along the river and the 
stage. The other essential quantity, the cross-section area, also as a 
function of location along the river and of stage, could in principle 
be determined from contour maps of the river valley; this is, in fact, 
the method used in building models, and it could have been used in 
setting up the problem for numerical calculation in the manner under 
discussion here. If that had been done, the results obtained would 
probably have been more accurate; however, such a procedure is 
extremely laborious and time consuming, and since the other equally 
important empirical element, i.e. the resistance coefficient, is known 
only as an average over each of the reaches (this applies equally to 
the models of a river), it seems reasonable to make use of an average 
cross-section area over each reach also. Such an average cross-section 
area was obtained by analyzing data from past floods in such a way 
as to determine the water storage volumes in each reach, and from them 
an average cross-section area as a function of the river stages was 
calculated. In this way the coefficients of the differential equations are 
obtained as numerically tabulated functions of x and y. (It might 
perhaps be reasonable to remark at this point that the carrying out 
of this program is a fairly heavy task, which requires close cooperation 
with the engineers who are familiar with the data and who understand 
also what is needed in order to operate with the differential equations). 
In Fig. 11.7.1 a diagrammatic sketch of the Ohio River between 
Wheeling and Cincinnati is shown, together with the reaches and 
observation stations at their ends. What we now have are resistance 
coefficients and cross-section areas that represent averages over any 
given reach. However, the reaches are too long to serve as intervals 
for the method of finite differences — which is basic for the numerical 
integration of the differential equations. Rather, an interval between 

* Each such interval is called a reach by those who work practically with river 
regulation problems. 



MATHEMATICAL HYDRAULICS 



501 



net points (in the staggered scheme described in the preceding section) 
of 10 miles was taken in order to obtain a sufficiently accurate approx- 
imation to the exact solution of the problem. A time interval of 



Pittsburgh 



Cmcinno t 




Moysville 



Huntington 
Fig. 11.7.1. Reaches in the Ohio 



9 minutes was used. Actually, calculations were first made using a 
5-mile interval along the river, but it was found on doubling the inter- 
val to 10 miles that no appreciable loss in accuracy resulted. 

To begin with, flood predictions for the 1945 flood were made, start- 
ing at a time when the river was low and the flow was practically a 
steady flow. Calculations were first made for a 36 hour period during 
which the flood was rising; as stated earlier, these were made using 
the measured inflows from tributaries, and the estimated run-off 
in the main valley. Upon comparison with the actual records, it was 
found that the predicted flood stages were systematically higher than 
the observed flood stages, and that the discrepancy increased steadily 
with increase in the time. It seemed reasonable to suppose that the 
error was probably due to an error in the resistance coefficient. Con- 
sequently a series of calculations was made on the UNIVAC in which 
this coefficient was varied in different ways; from these results, cor- 
rected coefficients were estimated for each one of the reaches. Actuallv 



502 WATER WAVES 

this was done rather roughly, with no attempt to make corrections 
that would require a modification in the shape of these curves in their 
dependence on the stage. The new coefficients, thus corrected on the 
basis of 36-hour predictions (and thus for flood stages far under the 
maximum), were then used to make predictions for various 6-day 
periods, as well as some 16-day periods, with quite good results, on 
the whole. 

It might be said at this point that making such a correction of the 
resistance ocefficient on the basis of a comparison with an actual flood 
corresponds exactly to what is done in making model studies. There, 
it is always necessary to make a number of verification runs after the 
model is built in order to compare the observed floods in the model 
with actual floods. In doing so, the first run is normally made without 
making any effort to have the resistance correct — in fact, the rough- 
ness of the concrete of the model furnishes the only resistance at the 
start. Of course it is then observed that the flood stages are too low 
because the water runs off too fast. Brass knobs are then screwed 
into the bed of the model, and wire screen is placed at some parts of 
the model, until it is found that the flood stages given by the model 
agree with the observations. This is, in effect, what was done in 
making numerical calculations. In other words, the resistance cannot 
be scaled properly in a model, but must be taken care of in an empi- 
rical way. The model is thus not a true model, but, as was stated earlier, 
it is rather a calculating machine of the class called analogue com- 
puters. It is, however, a very expensive calculating machine which can. 
in addition, solve only one very restricted problem. A model of two 
fair sized rivers, for example, consisting of two branches perhaps 200 
miles in length upwards from their junction, together with a short 
portion below the junctions, could cost more than a UNIVAC. 

It has already been stated that average cross-section areas for the 
individual reaches were used in making the numerical computations, 
while in the model the cross-sections are obtained from the contour 
maps. In operating numerically it is possible to change the local cross- 
section areas without any difficulty, and this might be necessary at 
certain places along the river. 

Some idea of the results of the calculations for the 1945 flood in the 
Ohio is given by Fig. 11.7.2. The graph shows the river stage at Po- 
meroy as a function of the time. At the other stations the results were 
on the whole more accurate. The graph marked "computation with 
original data", and which covers a 36 hour period, was computed on 



MATHEMATICAL HYDRAULICS 



503 



the basis of the resistance coefficients as estimated from the basic 
flow data for the river. As one sees, these coefficients resulted in much 
too high stages, and corrections to them were made along the river 




Fig. 11.7.2. Comparison of calculated with observed stages at Pomeroy for the 
1945 flood in the Ohio River 



on the basis of the results of this computation. Afterwards, flood 
predictions were made for periods up to 16 days without further 



Thebes 



mississipp 



Metropoli s 




JUNCTION 



32mi 



Hickman * s 

Fig. 11.7.3. The junction of the Ohio and the Mississippi 



504 



WATER WAVES 



correction of these coefficients. The graph indicates results for a 6 day 
period during which the flood was rising. Evidently, the calculated 
and observed stages agree very well. 



300 



296- 



292 



288 




Jan 15 18 21 24 27 30 



312 



308 



304 



300 




Observed stages 
Computed hydrograph 



Stage at Cairo 



Jan 15 18 21 24 27 30 

Fig. 11.7.4. Calculated and observed stages at Cairo and Hickman 

In Fig. 11.7.3 a diagrammatic sketch of the junction of the Ohio 
and the Mississippi is shown indicating the portions of these rivers 
which entered into the calculation of a flood coming down the Ohio 
and passing through the junction. The flood in question was that of 



MATHEMATICAL HYDRAULICS 505 

1947. It was assumed that the stages at Metropolis in the Ohio (about 
40 miles above Cairo) and at Thebes in the upper Mississippi (also 
about 40 miles above Cairo) were given as a function of the time. At 
Hickman in the lower Mississippi (about 40 miles below Cairo) the 
stage-discharge relation at this point, as known from observations, 
was used as a boundary condition. The results of a calculation for a 
16 day period are shown in Fig. 11.7.4, which gives the stages at 
Cairo, and at the terminating point in the lower Mississippi, i.e. at 
Hickman. As one sees, the accuracy of the prediction is very high, 
the error never exceeding 0.6 foot. It might be mentioned that a 
prediction for 6 days requires about one hour of calculating time 
on the UNIVAC, so that the calculating time for the 16 day period 
was under 3 hours, which seems reasonable. This problem of rout- 
ing a flood through a junction is, as has been mentioned before, 
one which has not been dealt with successfully by the engineering 
methods used for flood routing in long rivers.* 



Appendix to Chapter 11 



Expansion in the neighborhood of the first characteristic 

It has been mentioned already that whereas the forerunner of a 
disturbance initiated at a certain point in a river at a moment when 
the flow is uniform travels downstream with the speed v + Vgy> the 
main part of the flood wave travels more slowly (cf. Deymie [D.9] ), 
depending strongly on the resistance of the river bed. An investigation 
of the motion near the head of the wave, i.e. near the first characteris- 
tic (cf. the first part of sec. 11.6) with the equation x = (v + c )t. 
shows immediately why the main part of the disturbance will in 
general fall behind the forerunners of the wave. 

The motion is investigated in this Appendix by means of an ex- 
pansion in terms of a parameter that has been devised by G. Whitham 
and A. Troesch and carried out to terms of the two first orders for the 
model of the Ohio River, and to the lowest order in the much more 

* Added in proof: In the meantime, calculations have been completed (see [1. 4a] ) 
for the case of floods through the Kentucky Reservoir at the mouth of the 
Tennessee River. The calculated and observed stages differed only by inches for 
a flood period of three weeks over the 186 miles of the resevoir. 



506 



WATER WAVES 



complicated case of the junction problem. The results obtained make 
it possible to improve the accuracy of the solution near the first 
characteristic which separates the region of undisturbed flow from 
that of the flood wave. It turns out that the finite difference scheme 
yields river depths which are too large, as indicated by Fig. ll.A.l. 



profile computed by 
^v finite differences 




Fig. ll.A.l. Error introduced by finite difference scheme in neighborhood of 
first characteristic of a rapidly rising flood wave 



In order to expand the solution in the neighborhood of the wave 
front, we introduce new coordinates £ and % as follows: 

| = x and r = (v + c )t — x 

such that the |-axis (i.e. t = 0) coincides with the first characteristic. 
Near the front of the wave r will be small, and the expansion will be 
carried out by developing v and c in powers of r. The basic system of 
equations is restated for convenience: 

2cc x + v t + vv x - gS + gS f = 0, 
cv x + 2vc x + 2c t = 0. 

Upon substitution of the new variables | and r we find 

2c(c f -c T ) + v{v $ -v r ) + (u + c )v r - gS + gS f = 0, 



ll.A.l 



2v(c { 



+ c(Vi 



+ 2(v + c )c T 



where the friction slope S f for a rectangular channel of width B is given 

by 



Sf 



v\v\ 



,4/3 



*(■+!)) 



V\V\ 



1 2 

- + 



gB 



4/3 



MATHEMATICAL HYDRAULICS 



507 



We expand v and c as power series in x with coefficients that are func- 
tions of £ as follows: 

V = V Q + »i(f )t + M^)? 2 + • • ■» 
C = C + Cj(f )t + c 2 (£)t 2 + 

This expansion is to be used for r > only, since for t < we are in 
the undisturbed region and all the functions v 1 (i) i v 2 (£), . . ., c x (f ), 
c 2 (£)> . . . vanish identical^. If we insert the series for v and c into 
equations (ll.A.l) and collect terms of the same order in r, we get 
ordinary differential equations for v x (^), c^tj), .... The equations 
resulting from the terms of zero order in t yield v 1 = 2c 1 . The 
first order terms become, after thus eliminating v 1 , 



(11.A.2) 



dc, (1 2 

2(w + c o)~^ ~ 6cJ + 2c z; 2 - 4c c 2 + 4 Cl g£ - - — 



1 + 



2(i> +*o) 






2 < 



6c — 2c v 2 + 4c c 2 



By adding these two equations and removing the common factor 4, 
we find the differential equation for c x (£) is: 

dc 1 
~dk 



(Vq + Co)* 



3cf + Cl gS 



2 

3c, 



1 + 






0. 



Although the solution of this differential equation for c^) is easily 
obtained, the result expressed in general terms is complicated, and it 
is preferable to give it only for the case of the model of the Ohio River 



wove front 




0. 



Fig. 11. A. 2. Behavior near the front of a wave 



508 



WATER WAVES 



using the parameter values introduced above. In this case we find: 
Cl = (1.05 + 8.06 e oim )-\ with c 1 and | in miles and hours. This re- 
sult has the following physical meaning: The angle a of the profile 
measured between the wave front and the undisturbed water surface 
dies out exponentially: a < — '1/(1 + ae bx ), with a and b constants de- 
pending on the river and the boundary condition at x = 0. Theore- 
tically, a could also increase exponentially downstream so that a bore 
would eventually develop, but only if the increase in level at x = is 
extremely fast; in our example no bore will develop unless the water 
rises at the extremely rapid rate of at least 1 ft per minute. 

Unfortunately, the evaluation of c 2 (£), which yields the curvature of 



t ; 


i 






hours 








10 


- 




^-x - 1 9. 7 1 


5 










i 
50 


100 


150 x , 



miles 



Fig. 11. A. 3. Region of practically undisturbed flow 



the profile at the wave front, is already very cumbersome. The curva- 
ture is found to decrease for large x like xe~ bx , b being a positive con- 
stant. With the two highest order terms in the expansion known, it is 
possible to estimate the region adjacent to the first characteristic 
where the flow is practically undisturbed. It is remarkable how far 
behind the forerunner the first measurable disturbance travels (see 
Fig. 11.A.3). 

In a similar way, an expansion as a power series in x has been carried 
out for the problem of the junction of the Ohio and Mississippi, as 
described in earlier sections. Here even the lowest order term was 
obtained only after a complicated computation, since it was necessary 
to work simultaneously in three different x, ^-planes, with boundary 
conditions at the junction. The differential equations for c x are. in all 
three branches, of the same type as for the Ohio, and their solution 
for the junction problem with the parameters of section 11.6 



MATHEMATICAL HYDRAULICS 509 

are c x = .00084 t ,1451 for the upstream branch of the Mississippi, 
and c 1 = .00084 e - - 2291 for the downstream branch of the Mississippi, 
c x and £ both being given in miles and hours. This means that the 
angle a also dies out exponentially in the Mississippi, a little faster 
downstream than upstream, as might have been expected, since the 
oncoming water in the upstream branch has the affect of making the 
wave front steeper. 

In the problem of the idealized Ohio River and of the idealized 
problem of its junction with the Mississippi River the expansions 
were carried out numerically in full detail and were used to avoid 
computation by finite differences in a region of practically undisturb- 
ed flow.* 



* This would become more and more important if the flow were to be com- 
puted beyond 10 hours. 



PART IV 



CHAPTER 12 

Problems in which Free Surface Conditions are Satisfied 
Exactly. The Breaking of a Dam. Levi-Civita's Theory 

This concluding chapter constitutes Part IV of the book. In Part I 
the basic general theory and the two principal approximate theories 
were derived. Part II deals with problems treated by means of the 
linearized theory arising from the assumption that the motion is a 
small deviation from a state of rest or from a uniform flow. Part III 
is concerned with the approximate nonlinear theory which arises 
when the depth of the water is small, but the amplitude of the waves 
need not be small. Finally, in this chapter we deal with a few problems 
in which no assumptions other than those involved in the basic general 
theory are made. In particular, the nonlinear free surface conditions 
are satisfied exactly. 

The first type of problem considered in this chapter belongs in the 
category of problems concerned with motions in their early stages 
after initial impulses have been applied. A typical example is the 
motion of the water in a dam when the dam is suddenly broken. This 
problem will be treated along lines worked out by Pohle [P.ll], 
[P. 12]. Similar problems involving the collapse of a column of liquid 
in the form of a circular half-cylinder or of a hemisphere resting on a 
rigid bottom have been treated by Penney and Thornhill [P. 2] by 
a method different from that used by Pohle. 

The second section of the chapter deals with the theory of steady 
progressing waves of finite amplitude. The existence of exact solutions 
of this type is proved, following in the main the theory worked out 
by Levi-Civita [L.7]. 

12.1. Motion of water due to breaking of a dam, and related problems 

With the exception of the present section we employ throughout 
this book the so-called Euler representation in which the velocity and 
pressure fields are determined as functions of the space variables and 

513 



514 WATER WAVES 

the time. In this section it is convenient to make use of what is com- 
monly called the Lagrange representation, in which the displacements 
of the individual fluid particles are determined with respect to the 
time and to parameters which serve to identify the particles. Usually 
the parameters used to specify individual particles are the initial 
positions of the particles, and we shall conform here to that practice. 
Only a two-dimensional problem will be treated in detail here; con- 
sequently we choose the quantities a, b, and t as independent variables, 
with a and b representing Cartesian coordinates of the initial positions 
of the particles at the time t = 0. The displacements of the particles 
are denoted by X(a, b; t) and F(a, b; t), and the pressure by p(a, b; t). 
The equations of motion are 

Xtt = - - Vx 
Q 

Y tt = - -Vy - g 
Q 

in accord with Newton's second law. We assume gravity to be the 
only external force. These equations are somewhat peculiar because 
of the fact that derivatives of the pressure p with respect to the de- 
pendent variables X and Y occur. To eliminate them we multiply by 
X a and Y a , respectively, and add, then also by X b , Y b , and add; 
the result is 



(12.1.1 



X tt X a + (Y ti + g)Y a + l p a = 0, 

Q 

X tt X b + (Y tt + g)Y b +l Pb = 0, 

Q 



and these are the equations of motion in the Lagrangian form. These 
equations are not often used because the nonlinearities occur in an 
awkward way; however, they have the great advantage that a solu- 
tion is to be found in a fixed domain of the a, 6-plane even though 
a free surface exists. For an incompressible fluid — the only case 
considered here -the continuity condition is expressed by requiring 
that the Jacobian of X and Y with respect to a and b should remain 
unchanged during the flow (since an area element composed always 
of the same particles has this property); but since X = a and Y =- b 
initially, it follows that 

(12.1.2) X.Y t - X b Y„ = 1 



THE BREAKING OF A DAM 515 

is the condition of continuity. If the pressure p is eliminated from 
(12.1.1) by differentiation the result is 

(12.1.3) (X a X bt + Y a Y M ) t = (X b X at + Y b Y at ) t . 
Integration with respect to t leads to 

(12.1.4) (U, + Y a Y bl ) - (X b X al + Y„Y al ) = /(«, b) 

with / an arbitrary function. It can easily be shown by a calculation 
using the Eulerian representation that the left hand side of this equa- 
tion represents the vorticity; consequently the equation is a verifi- 
cation of the law of conservation of vorticity. If the fluid starts from 
rest, or from any other state with vanishing vorticity, the function 
f(a, b) would be zero. 

The method used by Pohle [P.ll], [P. 12] to solve the equations 
(12.1.1) and (12.1.2) — which furnish the necessary three equations for 
the three functions X, Y, and p — consists in assuming that solutions 
exist in the form of power series developments in the time, with co- 
efficients which depend on a and b: 



(12.1.5 



X(a, b;t) = a + X™(a, b) ■ t + X< 2 >(«, b) • t 2 + . . ., 
Y{a, b;t) =b + F (1) («, b) • t + F< 2 >(«, b) • t 2 + . . ., 
I p(a, b; t) = p (Q) (a, b) + p (1) (a, b) • t + p (2) {a, b) • t 2 + 



In these expansions we observe that the terms of order zero in X and Y 
are a and 6 — in accordance with the basic assumption that these 
quantities fix the initial positions of the particles. It should also be 
noted that X {1) and F (1) are the components of the initial velocity, 
and X (2) and F (2) similarly for the acceleration; in general, we would 
therefore expect that X {1) and F (1) would be prescribed in advance as 
part of the initial conditions. Of course, boundary conditions imposed 
on X, F, and p would lead to boundary conditions for the coefficient 
functions in the series developments. The convergence of the series 
for the cases discussed below has not been studied, but it seems likely 
that the series would converge at least for sufficiently small values of 
the time. The convergence of developments of this kind in some simp- 
ler problems in hydrodynamics has been proved by Lichtenstein 
[L.12]. 

The series (12.1.5) are inserted first in equation (12.1.2) and the 
coefficient of each power of t is equated to zero with the following 
result for the first two terms: 



516 WATER WAVES 

I X^ 4- F< 1} — 
(12 16} a b ' 

K ' ' } \xp+ Ff = - (X&YP - XPYP). 

We observe that X {1) and F (1) are subject to the above relation and 
hence cannot both be prescribed arbitrarily; however, if the fluid 
starts from rest so that X {1) = F (1) = 0, the condition is automatic- 
ally satisfied. The equation for X i2) and F (2) is linear in these quanti- 
ties, but nonlinear in X {1) and F (1) . This would be the situation in 
general: X {n) and F (n) would satisfy an equation of the form 

X (n) + yip) = f(x<u, Y^\ X™, F< 2 >, . . ., X<»-« Y in ~ 1) ) 9 

with F a nonlinear function in X (i \ F (i) , i = 1, 2, . . ., n — 1. In 
the following we shall consider only motions starting from rest. Con- 
sequently, we have X {1) = F (1) = 0, and equation (12.1.4) holds 
with / = 0; a substitution of the series in powers of t in equation 
(12.1.4) yields (for the lowest order term): 

(12.1.7) XJP - Ff = 0. 

The higher order coefficients satisfy an equation of the form 
X (n) _ Y (n) = G ( X W, Y^, . . ., Xi n -v, Y^- 1 )), with G a nonlinear 
function of X {i) , Y™, i = 2, 3, . . ., n — 1. Thus we observe that X (2) 
and F (2) satisfy the Cauchy-Riemann equations and are therefore 
conjugate harmonic functions of a and b. The higher order coefficients 
would satisfy Poisson's equation with a right hand side a known 
function fixed by the coefficient functions of lower order. Thus the 
coefficients in the series for X and F can be determined step-wise by 
solving a sequence of Poisson equations. Once the functions X (i) and 
Y (i) have been determined, the coefficients in the series for the 
pressure p can also be determined successively by solving a sequence 
of Poisson equations. To this end we of course make use of equations 
(12.1.1); the result for p (o >(a, b) is 

(12.1.8) p(j) + pio) = - 2q(X® + Ff ) = 0, 

from (12.1.6) and X (1) = F (1) = 0. Thus p {0) (a, b) is a harmonic 
function. For p {n) (a, b) one would find a Poisson equation with a right 
hand side determined by X (i) and Y (i) for i = 2, 3, . . ., n + 2. 

It would be possible to consider boundary conditions in a general 
way, but such a procedure would not be very useful because of its 
complexity. Instead, we proceed to formulate boundary conditions 
for the special problem of breaking of a dam, which is in any case 
typical for the type of problems for which the present procedure is 



THE BREAKING OF A DAM 



517 



recommended. We assume therefore that the region occupied initially 
by the water (or rather, a vertical plane section of that region) is the 
half-strip <^ a < oo, ^ b ^ h, as indicated in Fig. 12.1.1. The 




a=0 

b=0 o 

Fig. 12.1.1. The breaking of a dam 



dam is of course located at a = 0. Since we assume that the water 
is initially at rest when filling the half-strip we have the conditions 



(12.1.9) 

and 

(12.1.10 



X(a, b; 0) = a, Y(a, b; 



X t (a, b; 0) = 0. 



Y t (a, b; 0) = 0. 



When the dam is broken, the pressure along it will be changed 
suddenly from hydrostatic pressure to zero; it will of course be pre- 
scribed to be zero on the free surface. This leads to the following 
boundary conditions for the pressure: 



(12.1.11 



jp(a,h;t) = 0, 
\p(0,b;t) = 0, 



^ a < oo, 
^ b ^ h, 



t >0, 

t > 0. 



Filially the boundary condition at the bottom b = results from the 
assumption that the water particles originally at the bottom remain 
in contact with it; as a result we have the boundary condition 

(12.1.12) Y(a,0;t) = 0, ^ a < oo, t > 0. 

The conditions (12.1.9) are automatically satisfied because of the 
form (cf. (12.1.5)) chosen for the series expansion. The conditions 
(12.1.10) are satisfied by taking X (1) (a, b) = F (1) (a, b) = 0. 

In order to determine the functions X {2) (a, b) and F (2) (a, b), it is 
necessary to obtain boundary conditions in addition to the differential 
equations given for them by (12.1.6) and (12.1.7). Such boundary 
conditions can be obtained by using (12.1.11) and (12.1.12) in con- 
junction with (12.1.1) and the power series developments. Thus from 



518 WATER WAVES 

(12.1.12) we find Y (2) (a, 0) = for ^ a < oo (indeed, Y {n) (a, 0) 
would be zero for all ?i). Insertion of the series (12.1.5) and use of the 
boundary conditions for b = h yields 

(12.1.13) X^(a, h) = 0, 

upon using the first of the equations in (12.1.1). The second equation 
of (12.1.1) leads to the condition 

(12.1.14) F< 2 >(0, b) = - ?. 

We know that Z(z) = Y (2) -\- iX (2) is an analytic function of the 
complex variable ; = a -f- ib in the half-strip, and we now have 
prescribed values for either its real or its imaginary part on each of the 
three sides of the strip; it follows that the function Z can be deter- 
mined by standard methods— for example by mapping conformally 
on a half plane. In fact, the solution can be given in closed form, as 
follows: Since X i2) (a, h) = 0, we see that X {2) (a, h) = 0, and hence 
that Y {2) (a, h) = since X {2) and Y {2) are harmonic conjugates. 
Therefore the harmonic function Y {2) (a, b) can be continued over the 
line b = h by reflection into a strip of width 2h, as indicated in Fig. 
12.1.2; the boundary values for Y (2) are also shown. Thus a complete- 



rs) g 

T " 2 



b = 2h 



b = h 



b=0 



Fig. 12.1.2. Boundary value problem for Y (2) (a, b) 

ly formulated boundary value problem for Y (2) (a, 6) in a half-strip 
has been derived. To solve this problem we map the half-strip on the 
upper half of a w-plane by means of the function w = cosh (7izj2h) 
— either by inspection or by using the Schwarz-Christoffel mapping 
formula — and observe that the vertices z = and z = 2ih of the half- 
strip map into the points w = ± 1 of the w-plane, as indicated in 
Fig. 12.1.3. The appropriate boundary values for Y (2) (w) on the real 
axis of the w-plane are indicated. The solution for Y {2) (w) under 



THE BREAKING OF A DAM 



519 




o-i _y. +i o 

2 

Fig. 12.1.3. Mapping on the w-plane 



these conditions is well known; it is the function Y {2) (P) = 
— (g/27i)(0 2 — 0i), with d 1 and d 2 the angles marked in Fig. 12.1.3. 
The analytic function of which this is the real part is well known; it is 

it* w — 1 
y<2> + ixw = - J- log 



2tt 



W> + 1 



as can in any case be easily verified. Transferring back to the z-plane 
we have 



Z(z) 



y( 2 ) + ix m = - -A log 

2n * 



. 7ZZ 

cosh — 
2h 



cosh — + 1 
2h 



and upon separation into real and imaginary parts we have finally: 



(12.1.15) 



X<V{a, b) = - *- log 

2n 



2 nb . U2 m 

cos' 2 — + sinrr — 

Ml 4h 

Tib . TTtt 

sin- 5 — 4- sinh/ — 
4& 4/i 



F< 2 >(a, &) = 



arctan 



sin 



nb 
2h 



sinh 



na 
2h 



One checks easily that the boundary conditions X (2) (a, h) = 0, 
F (2) (a, 0) = are satisfied, and that F (2) (0, b) = — g/2. The initial 
pressure distribution p (0) (a, 6) can be calculated, now that X {2) (a, b) 
is known, by using the first equation of (12.1.1), which yields 



520 WATER WAVES 

(12.1 16) p { a o} = - 2qXW. 

In the present case there are advantages in working first with the 
pressure p(a, 6; t) and determining the coefficient of the series for it 
directly by solving appropriate boundary value problems; afterwards 
the coefficients of the series for X and Y are easily found. The main 
reason for basing the calculation on the pressure in the first instance 
is that the boundary conditions at 6 = h and a = are very simple, 
i.e. p = and hence p {i) = for all indices i. The boundary conditions 
at the bottom 6 = involve the displacements Y. For instance, one 
finds readily in the same general way as above that p^ = — gg, 
p^ = 0, and pf} = — QgYj® as boundary conditions at 6 = 0. 
Since p {0) is harmonic, it is found at once without reference to dis- 
placements—an interesting fact in itself. Once p io) is found, X (2) and 
Y (2) can be calculated without integrations (cf. (12.1.16), for example). 
Since p[ l) = for 6 = 0, and p {1) is also harmonic, it follows that 
p {1) (a, 6) = 0. Since Y (2) is now known, it follows that a complete set 
of boundary conditions for p {2) (a, 6) is known, and p {2) (a, b) is then 
determined by solving the differential equation 



(12.1.17) vy 2 > =p<g +p<?> 



bb 

d(X<*K F (2) ) 
d(a, b)~ 



V 2 {(X<*>) 2 ^ (F< 2 >) 2 +gF< 2 >} 



h 2 I cosh — 
I h 



na nb\ 

cos — | 
h 



whose right hand side is obtained after a certain amount of mani- 
pulation. This process can be continued. One would find next that 
XW = y<3) = o, and that X< 4 > and F< 4 > can be found once p< 2 > is 
known. However, the boundary condition at the bottom, and the right 
hand sides in the Poisson equations for the functions p {i) (a,b) become 
more and more complicated. 

The initial pressure p {o) (a, 6) can be discussed more easily on the 
basis of a Fourier series representation than from the solution in 
closed form obtainable from (12.1.16); this representation is 

(12.1.18) p {0) {a, 6) 

(2n +l)nb 



Spgh 


00 

I 

n=0 




1 




(2n + l)na 








— e 


2h 


71* 


(2rc 


+ 


i)« 





= Qg{h - 6) - -£— y — — - e- 2h cos 



2h 



THE BREAKING OF A DAM 



521 



We note that the first term represents the hydrostatic pressure, and 
that the deviation from hydrostatic pressure dies out exponentially 
as a -> cc and also as h -> 0, i.e. on going far away from the dam and 
also on considering the water behind the dam to be shallow (or, better, 
considering ajh to be large). This is at least some slight evidence of 
the validity of the shallow water theory used in Chapter 10 to discuss 
this same problem of the breaking of a dam — at least at points not 
too close to the site of the dam. 

The shape of the free surface of the water can be obtained for small 
times from the equations 



(12.1.19) 



X = a + X^t\ 



b + Y< 2 H 2 



evaluated for a = (for the particles at the face of the dam) and for 
b = h on the upper free surface. The results of such a calculation for 
the specific case of a dam 200 feet high are shown in Fig. 12.1.4. 



h ; 

feet 


200 






-150 




t =i.5>y/ 


-100 




r^ — i^t — i — 


-50 


1 >■ 



50 40 30 20 10 



100 



X 

miles 



Fig. 12.1.4. Free water surface after the breaking of a dam 



One of the peculiarities of the solution is a singularity at the origin 
a = 0, 6 = which is brought about by the discontinuity in the 
pressure there. In fact, X {2) has a logarithmic singularity for a = 0, 
b = 0, as one sees from (12.1.15) and X is negative infinite for all 
t =£ 0. This, of course, indicates that the approximation is not good 
at this point; in fact, there would be turbulence and continuous 
breaking at the front of the wave anyway so that any solution 
ignoring these factors would be unrealistic for that part of the flow. 
In the thesis by Pohle [P.ll], the solution of the problem of the 
collapse of a liquid half-cylinder and of a hemisphere on a rigid plane 
are treated by essentially the same method as has been explained for 



522 WATER WAVES 

the problem of the breaking of a dam. These problems have also been 
treated by Penney and Thornhill [P.2], who also use power series in 
the time but work with the Eulerian rather than the Lagrangian re- 
presentation, which leads to what seem to the author to be more com- 
plicated calculations than are needed when the Lagrangian represen- 
tation is used. 

12.2. The existence of periodic waves of finite amplitude 

In this section a proof, in detail, of the existence of two-dimensional 
periodic progressing waves of finite amplitude in water of infinite 
depth will be given. This problem was first solved by Nekrassov 
[N.l, la] and later independently by Levi-Civita [L.7]; Struik 
[S.29] extended the proof of Levi-Civita to the same problem for 
water of finite constant depth. A generalization of the same theory 
to liquids of variable density has been given by Dubreuil-Jacotin 
[D.15, 15a]. Lichtenstein [L.ll] has given a different method of 
solution based on E. Schmidt's theory of nonlinear integral equations. 
Davies [D.5] has considered the problem from still a different point of 
view. Gerber [0.5] has recently derived theorems on steady flows in 
water of variable depth by making use of the Schauder-Leray theory. 

We shall start from the formulation of the problem given by Levi- 
Civita (and already derived in 10.9 above), but, instead of proving 
directly, as he does, the convergence of a power series in the ampli- 
tude to the solution of the problem, an iteration procedure devised 
by W. Littman and L. Nirenberg will be used to establish the existence 
of the solution. The two procedures are not, however, essentially 
different. 

It is convenient to break up this rather long section into sub-sec- 
tions as a means of focusing attention on separate phases of the 
existence proof. 

12.2a. Formulation of the problem 

As in sec. 10.9, the problem of treating a progressing wave which 
moves unchanged in form and with constant velocity is reduced to a 
problem of steady flow by observing the motion from a coordinate 
system which moves with the wave. A complex velocity potential (see 
sec. 10.9 for details) %(z) is therefore to be found in the as, ?/-plane 
(cf. Fig. 12.2.1): 



LEVI-CIVITA S THEORY 



523 



< h > 

u ' 



Fig. 12.2.1. Periodic waves of finite amplitude 



(12.2.2) 



w 



= u 



w 



(12.2.1) x = <P + ty = X( z )< z = x + iy. 

The velocity at y = — oo should be U. The real harmonic functions 
(p(x, y) and yj(x, y) represent the velocity potential and the stream 
function. The complex velocity w is given by 

dz 

with u, v the velocity components. This follows at once from the 
Cauchy-Riemann equations: 

(12.2.3) tp x = ip y = u, cp y = — y) x = v, 

since w = q> x + iip x . 

We proceed to formulate the boundary conditions at the free sur- 
face. The kinematic free surface condition can be expressed easily be- 
cause the free surface is a stream line, and we may choose ip(x, y) = 
along it. The dynamic condition expressed in Bernoulli's law is given 

by 

(12.2.4) J | w | 2 + gy = const. at xp = 0, 

as one can readily verify. The problem of satisfying this nonlinear 
condition is of course the source of the difficulties in deriving an 
existence proof. At oo the boundary condition is 

(12.2.5) w -> U uniformly as y -> — oo, 

and w is in addition supposed to be nowhere zero and to be uniformly 
bounded. We seek waves which are periodic in the .r-coordinate and 
thus we require i to satisfy the condition 

(12.2.6) X (z +h)- X (z) = 0, 
with h a real constant. 



524 WATER WAVES 

Following Levi-Civita, we assume that the region of flow in the 
z-plane is mapped into the cp, ^-plane by means of %(z). The free sur- 
face in the physical plane corresponds to the real axis tp = of the 
^-plane, and we assume that the entire region of the flow in the 2-plane 
is mapped in a one-to-one way on the lower half of the ^-plane. (We 
shall prove shortly that a function %(z) satisfying the conditions given 
above would have this property.) In this case the inverse mapping 
z(%) exists, and we may regard the complex velocity w(z) as an ana- 
lytic function of % defined in the lower half of the ^-plane. In this 
way we are enabled to work with a domain in the <p, ^-plane that is 
fixed in advance instead of with an unknown domain of the x, y -plane. 
Levi-Civita goes a step further by introducing a new dependent varia- 
ble co, replacing w, by the relation 

(12.2.7) w = Ue- im , co = 6 + ir; 

so that co is an analytic function of cp + iip. Consequently we have 
(cf. (12.2.2)) 

(12.2.8) | w | = Ue r , d = argw. 

Thus r = log (\w\/U), while 6 is the inclination of the velocity vector. 
In the same way as in sec. 10.9 (cf. the equations following (10.9.11)) 
the boundary condition (12.2.4) can be put in the form 

(12.2.9) V = AV- 3T sin 0, for tp = 0, 
with X' defined by 

(12.2.10) X' = g/U 3 . 

Our problem now is to determine an analytic function co(%) = 
0(cp,ip) + ir((p, ip) in the lower halfplane \p < and a constant X' in 
(12.2.9) such that a) co is analytic for tp < 0, continuous for \p ^ 0, 
b) d w is continuous for tp ^ and the nonlinear boundary condition 
(12.2.9) is satisfied, c) co has the period Uh in cp, d) co{%) -> as 
tp ->— oo, e) \ co(x) \ ^2- The last two conditions